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Trigonometric Equation Calculator

Instructions: Use calculator to solve trigonometric equations you provide, showing all the steps. Please type in the trigonometric equation you want to carry out in the box below.

ABout this Trigonometric Equation Calculator

This calculator will allow you to solve trig equations, showing all the steps of the way. All you need to do is to provide a valid trigonometric equation, with an unknown (x). It could be something simple as 'sin(x) = 1/2', or something more complex like 'sin^2(x) = cos(x) + tan(x)'.

Once you are done typing your equation, just go ahead and click on "Solve" to get all the details of the processes of finding the solutions, if solutions can be found.

Trigonometric properties and rules almost always allow to reduce most trig equations into simpler ones, so this type of equation is one type that often times lead to solutions, but it can be extremely cumbersome at times.

What is a trigonometric Equation?

A trigonometric equation, in the simplest possible terms, is a math equation where the unknown x is inside of a trigonometric expression.

For example, the following expression is a trigonometric eqn:

Why? Simply because x appears inside of trig expression sine. Or for example:

Now, these two are trig eqns, but the difference between the two is that for the first one, x appears ONLY inside of sine, whereas in the second one x appears inside of a trig function (tangent), but it also appears outside. This will usually make it difficult (or impossible) to solve the equation.

How to solve trigonometric equations

• Step 1: Make sure you are dealing with a trigonometric equation. Non-trigonometric equations will likely require a different approach
• Step 2: Make sure that the unknown x is inside of the trigonometric expression , but x does not appear outside a trig expression. If that is the case, it is likely you won't be able to solve the equation with elementary methods
• Step 3: Conduct an appropriate substitution, by first expressing all the trig functions present in the equation into one type (typically sine), and then use a substitution involving sine
• Step 4: With a little luck and if you did the correct substitution, you have reduced the original trigonometric equation into a polynomial equation to solve .

One of the key trig rules you need to use the ability to express all trigonometric functions in terms of any fixed trigonometric function. For example, we can write cosine in terms of sine:

Trigonometric Substitutions

Using trigonometric identities and substitutions is your way to go in this case. For example, suppose you want to solve this:

So we know this is a trig equation, and we know we can write cosine in terms of sine, so we do this:

Now what? Well, we can use the substitution: \(u = \sin x\), so the equation above becomes:

which is a rational equation , which by using simple algebraic manipulation means that we need to solve a polynomial equation in order to solve the original trig equation.

Application of trigonometry

• Step 1: All things mechanical: In manufacturing mechanical parts, circles and trigonometry play a crucial role
• Step 2: Analysis of periodic functions: Many phenomena are tightly related to periodicity, the point at which trigonometry comes to play
• Step 3: Advanced math: Mathematicians love their Fourier Series and Transform, which play a crucial role in spectral analysis

Circles and all their symmetry are so really important in real life, and trigonometry is the language by which we can quantify circles and its relationships. Solving trigonometric equations is at the center of math.

Why would you solve trigonometric equations

Trigonometric equations carry lots of value in practice especially in Engineering. Notable properties such as the period and frequency open a full spectrum of applications.

Circular structures play a crucial role in everything mechanical that we use today. Circles are synonymous of trigonometry, and trigonometric equations are at the center of it.

Example: Solving simple trig equations

Solve: \(\sin(x) = \frac{1}{2}\)

We need to solve the following given trigonometric equation equation:

The following is obtained:

By direct application of the properties of the inverse trigonometric function \( \arcsin(\cdot)\), as well as the properties of the trigonometric function \( \sin\left(x\right)\), we obtain that

Therefore, solving for \(x\) for the given equation leads to the solutions \(x=\frac{5}{6}\pi{}+2\pi{}K_1,\,\,x=\frac{1}{6}\pi{}+2\pi{}K_2\), for \(K_1, K_2\) arbitrary integer constants.

More equation calculators

Our trigonometric equation with steps will come in handy when dealing with equations with specific structure. If you are unsure of the type of equation you are dealing with, you can use our general equation solver , which will figure out the structure of the given equation, and will find a suitable approach.

The main difficulty with solving equations that are not linear equation or polynomial equation is that there is not a specific route to follow, nor there is any guarantee you will find solutions.

Usually, the strategy consists of simplifying expressions as much as possible, and after doing that, it is usually nowhere land, where you need to try whatever feels suitable.

Naturally, the idea is to try to reduce the equation to a simpler equation, by using some kind of substitution and a multi-step process, where you first find solutions of an auxiliary solution, which gives you CANDIDATES to the original equation. You would like to solve a linear equation , or even a quadratic equation , but perhaps the reduction you get will be a bit less generous.

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Get detailed solutions to your math problems with our trigonometry step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of trigonometry. This solution was automatically generated by our smart calculator:

Starting from the left-hand side (LHS) of the identity

Applying the trigonometric identity: $\cot\left(\theta \right) = \frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$

 Intermediate steps

Multiplying the fraction by $\cos\left(x\right)$

When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

Combine and simplify all terms in the same fraction with common denominator $\sin\left(x\right)$

Apply the trigonometric identity: $1-\cos\left(\theta \right)^2$$=\sin\left(\theta \right)^2$

Simplify the fraction $\frac{\sin\left(x\right)^2}{\sin\left(x\right)}$ by $\sin\left(x\right)$

Since we have reached the expression of our goal, we have proven the identity

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Our math problem solver that lets you input a wide variety of trigonometry math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

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Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

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Trigonometric equation solver

This calculator can solve basic trigonometric equations such as: $\color{blue}{ \sin(x) = \frac{1}{2} }$ or $ \color{blue}{ \sqrt{2} \cos\left(-\frac{3x}{4}\right) - 1 = 0 } $.

The calculator will find exact or approximate solutions on custom range. Solution can be expressed either in radians or degrees.

sin cos tan cot ( · x ) =

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Trigonometry : Solving Word Problems with Trigonometry

Study concepts, example questions & explanations for trigonometry, all trigonometry resources, example questions, example question #1 : solving word problems with trigonometry.

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

You can draw the following right triangle from the information given by the question.

In order to find the height of the flagpole, you will need to use tangent.

You can draw the following right triangle from the information given in the question:

In order to find out how far up the ladder goes, you will need to use sine.

In right triangle ABC, where angle A measures 90 degrees, side AB measures 15 and side AC measures 36, what is the length of side BC?

This triangle cannot exist.

Example Question #5 : Solving Word Problems With Trigonometry

A support wire is anchored 10 meters up from the base of a flagpole, and the wire makes a 25 o angle with the ground. How long is the wire, w? Round your answer to two decimal places.

23.81 meters

28.31 meters

21.83 meters

To make sense of the problem, start by drawing a diagram. Label the angle of elevation as 25 o , the height between the ground and where the wire hits the flagpole as 10 meters, and our unknown, the length of the wire, as w. 

Now, we just need to solve for w using the information given in the diagram. We need to ask ourselves which parts of a triangle 10 and w are relative to our known angle of 25 o . 10 is opposite this angle, and w is the hypotenuse. Now, ask yourself which trig function(s) relate opposite and hypotenuse. There are two correct options: sine and cosecant. Using sine is probably the most common, but both options are detailed below.

We know that sine of a given angle is equal to the opposite divided by the hypotenuse, and cosecant of an angle is equal to the hypotenuse divided by the opposite (just the reciprocal of the sine function). Therefore:

To solve this problem instead using the cosecant function, we would get:

The reason that we got 23.7 here and 23.81 above is due to differences in rounding in the middle of the problem. 

Example Question #6 : Solving Word Problems With Trigonometry

When the sun is 22 o above the horizon, how long is the shadow cast by a building that is 60 meters high?

To solve this problem, first set up a diagram that shows all of the info given in the problem. 

Next, we need to interpret which side length corresponds to the shadow of the building, which is what the problem is asking us to find. Is it the hypotenuse, or the base of the triangle? Think about when you look at a shadow. When you see a shadow, you are seeing it on something else, like the ground, the sidewalk, or another object. We see the shadow on the ground, which corresponds to the base of our triangle, so that is what we'll be solving for. We'll call this base b.

Therefore the shadow cast by the building is 150 meters long.

If you got one of the incorrect answers, you may have used sine or cosine instead of tangent, or you may have used the tangent function but inverted the fraction (adjacent over opposite instead of opposite over adjacent.)

Example Question #7 : Solving Word Problems With Trigonometry

From the top of a lighthouse that sits 105 meters above the sea, the angle of depression of a boat is 19 o . How far from the boat is the top of the lighthouse?

423.18 meters

318.18 meters

36.15 meters

110.53 meters

To solve this problem, we need to create a diagram, but in order to create that diagram, we need to understand the vocabulary that is being used in this question. The following diagram clarifies the difference between an angle of depression (an angle that looks downward; relevant to our problem) and the angle of elevation (an angle that looks upward; relevant to other problems, but not this specific one.) Imagine that the top of the blue altitude line is the top of the lighthouse, the green line labelled GroundHorizon is sea level, and point B is where the boat is.

Merging together the given info and this diagram, we know that the angle of depression is 19 o  and and the altitude (blue line) is 105 meters. While the blue line is drawn on the left hand side in the diagram, we can assume is it is the same as the right hand side. Next, we need to think of the trig function that relates the given angle, the given side, and the side we want to solve for. The altitude or blue line is opposite the known angle, and we want to find the distance between the boat (point B) and the top of the lighthouse. That means that we want to determine the length of the hypotenuse, or red line labelled SlantRange. The sine function relates opposite and hypotenuse, so we'll use that here. We get:

Example Question #8 : Solving Word Problems With Trigonometry

Angelina just got a new car, and she wants to ride it to the top of a mountain and visit a lookout point. If she drives 4000 meters along a road that is inclined 22 o to the horizontal, how high above her starting point is she when she arrives at the lookout?

9.37 meters

1480 meters

3708.74 meters

10677.87 meters

1616.1 meters

As with other trig problems, begin with a sketch of a diagram of the given and sought after information.

Angelina and her car start at the bottom left of the diagram. The road she is driving on is the hypotenuse of our triangle, and the angle of the road relative to flat ground is 22 o . Because we want to find the change in height (also called elevation), we want to determine the difference between her ending and starting heights, which is labelled x in the diagram. Next, consider which trig function relates together an angle and the sides opposite and hypotenuse relative to it; the correct one is sine. Then, set up:

Therefore the change in height between Angelina's starting and ending points is 1480 meters. 

Example Question #9 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 50 feet apart. The shorter building is 40 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 48 o . How high is the taller building?

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

Example Question #10 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 80 feet apart. The shorter building is 55 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 32 o . How high is the taller building?

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Right Triangle Trigonometry Calculator

Basics of trigonometry, right triangles trigonometry calculations, example of right triangle trigonometry calculations with steps, more trigonometry and right triangles calculators (and not only).

The right triangle trigonometry calculator can help you with problems where angles and triangles meet: keep reading to find out:

• The basics of trigonometry;
• How to calculate a right triangle with trigonometry;
• A worked example of how to use trigonometry to calculate a right triangle with steps;

And much more!

Trigonometry is a branch of mathematics that relates angles to the length of specific segments . We identify multiple trigonometric functions: sine, cosine, and tangent, for example. They all take an angle as their argument, returning the measure of a length associated with the angle itself. Using a trigonometric circle , we can identify some of the trigonometric functions and their relationship with angles.

As you can see from the picture, sine and cosine equal the projection of the radius on the axis, while the tangent lies outside the circle. If you look closely, you can identify a right triangle using the elements we introduced above: let's discover the relationship between trigonometric functions and this shape.

Consider an acute angle in the trigonometric circle above: notice how you can build a right triangle where:

• The radius is the hypotenuse; and
• The sine and cosine are the catheti of the triangle.

α \alpha α is one of the acute angles, while the right angle lies at the intersection of the catheti (sine and cosine)

Let this sink in for a moment: the length of the cathetus opposite from the angle α \alpha α is its sine , sin ⁡ ( α ) \sin(\alpha) sin ( α ) ! You just found an easy and quick way to calculate the angles and sides of a right triangle using trigonometry.

The complete relationships between angles and sides of a right triangle need to contain a scaling factor, usually the radius (the hypotenuse). Identify the opposite and adjacent . We can then write:

By switching the roles of the legs, you can find the values of the trigonometric functions for the other angle.

Taking the inverse of the trigonometric functions , you can find the values of the acute angles in any right triangle.

Using the three equations above and a combination of sides, angles, or other quantities, you can solve any right triangle . The cases we implemented in our calculator are:

• Solving the triangle knowing two sides ;
• Solving the triangle knowing one angle and one side ; and
• Solving the triangle knowing the area and one side .

Take a right triangle with hypotenuse c = 5 c = 5 c = 5 and an angle α = 38 ° \alpha=38\degree α = 38° . Surprisingly enough, this is enough data to fully solve the right triangle! Follow these steps:

• Calculate the third angle: β = 90 ° − α \beta = 90\degree - \alpha β = 90° − α .
• sin ⁡ ( α ) = 0.61567 \sin(\alpha) = 0.61567 sin ( α ) = 0.61567 .
• o p p o s i t e = sin ⁡ ( α ) ⋅ h y p o t e n u s e = 0.61567 ⋅ 5 = 3.078 \mathrm{opposite} = \sin(\alpha)\cdot\mathrm{hypotenuse} = 0.61567 \cdot 5 = 3.078 opposite = sin ( α ) ⋅ hypotenuse = 0.61567 ⋅ 5 = 3.078 .
• a d j a c e n t = 0.788 ⋅ 5 = 3.94 \mathrm{adjacent} = 0.788\cdot 5 = 3.94 adjacent = 0.788 ⋅ 5 = 3.94 .

If you liked our right triangle trigonometry calculator, why not try our other related tools? Here they are:

• The trigonometry calculator ;
• The cosine triangle calculator ;
• The sine triangle calculator ;
• The trig triangle calculator ;
• The trig calculator ;
• The sine cosine tangent calculator ;
• The tangent ratio calculator ; and
• The tangent angle calculator .

How do I apply trigonometry to a right triangle?

To apply trigonometry to a right triangle, remember that sine and cosine correspond to the legs of a right triangle . To solve a right triangle using trigonometry:

• sin(α) = opposite/hypotenuse ; and
• cos(α) = adjacent/hypotenuse .
• By taking the inverse trigonometric functions , we can find the value of the angle α .
• You can repeat the procedure for the other angle.

What is the hypotenuse of a triangle with α = 30° and opposite leg a = 3?

The length of the hypotenuse is 6 . To find this result:

• Calculate the sine of α : sin(α) = sin(30°) = 1/2 .
• Apply the following formula: sin(α) = opposite/hypotenuse hypotenuse = opposite/sin(α) = 3 · 2 = 6 .

Can I apply right-triangle trigonometric rules in a non-right triangle?

Not directly: to apply the relationships between trigonometric functions and sides of a triangle, divide the shape alongside one of the heights lying inside it. This way, you can split the triangle into two right triangles and, with the right combination of data, solve it!

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•    Solve Trigonometry Functions Without a Calculator

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Solve trigonometry functions without a calculator.

• Chloe Daniel
• Published On: March 05 ,2024

Trigonometry is the branch of mathematics that deals with the study of angles and their numerical calculations. Calculators come in handy while solving trigonometric functions because of their in-built sin, cos, and tan functions. However, if you want to be a math wizard and exercise your brain, you can do this simple math without a calculator easily.

Before learning how to challenge yourself with interesting trigonometry exercises, let’s understand its basics.

Branches of Trigonometry

There are three sub-branches of trigonometry:

1. Plane Trigonometry

Plane Trigonometry deals with the two-dimensional plane of a triangle. It states that when enough sides and angles of triangles are known, the remaining can be calculated easily using trigonometric functions. 

The two important laws mathematicians use to find the unknown sides and angles of the triangle are the law of sines and the law of cosines. To maintain consistency, the angles of the triangles are named A, B, and C; whereas, the sides of the triangle are called a, b, and c.

2. Spherical Trigonometry

Spherical Trigonometry involves the study of spherical triangles, formed due to the intersection of three big circle arcs on the surface of the sphere. Spherical triangles are different from planar ones because the sum of their angles exceeds the sum of angles in a flat triangle. 

Functions of spherical triangles include the angles (A, B, and C) sides (a, b, and c), and the dihedral angles (α, β, and γ).

A list of common spherical trigonometry formulas is given below:

Basic Trigonometric Concepts

In a right-angled triangle, we have the following three sides:

Perpendicular — the vertical side beside the right angle i.e., a.

Hypotenuse — the longest side opposite to the right angle i.e., c.

Base — The adjacent side to the angle θ i.e., b.

1. Trigonometric Functions

Trigonometry introduces three primary functions – sine (sin), cosine (cos), and tangent (tan). These functions define the relationship between angles and the sides of a triangle.

Sine (sin θ) : In a right triangle with angle θ, the sine ratio is calculated as the ratio of the length of the side opposite the angle to the hypotenuse. Mathematically, it can be expressed as:

Sin (θ)= Opposite sideHypotenuse

Cosine (cos θ) : For the same right triangle, the cosine ratio is the ratio of the length of the adjacent side to the hypotenuse. The formula is:

Cos (θ)= Adjacent  sideHypotenuse

Tangent (tan θ) : Tangent is the ratio of the length of the side opposite the angle to the length of the adjacent side. The formula is:

Tan (θ)= Opposite sideAdjacent

Tip to Remember: SOH CAH TOA

Here, S stands for sine, C for cosine, and T for the tangent. You can remember this simple trick to recall the above formulas easily.

2. Angles and Measurements

Angles are measured in degrees and radians. It's like deciding how big a slice of pizza you want from a whole pizza (circle).

Degrees : One full circle is 360 degrees. So, an angle can be measured as θ degrees.

Radians : Radians are a different way to measure angles. One full circle is 2π radians. The conversion between degrees (°) and radians (rad) is given by:

Radians = (180) x Degrees

Trigonometric Tables

Trigonometric tables are like cheat sheets for trigonometry. They're pre-calculated tables of values for sine, cosine, and tangent functions at specific angles. These tables were incredibly handy in the days before calculators and provided quick access to trigonometric values for various angles.

Using these tables involves finding the angle you're interested in.  Here's how it works:

Sine (sin) : If you want to find the sine of an angle, locate the angle in the table, and read off the corresponding sine value.

Cosine (cos) : Similarly, for cosine, find the angle and read off the cosine value.

Tangent (tan) : To find the tangent, divide the sine value by the cosine value for the given angle.

For example, if you have an angle of 30 degrees, you'd find the row for 30 degrees in the sine column to get the sine value. Repeat this process for cosine and tangent.

Tips for Memorizing Common Angles:

While tables are convenient, memorizing values for common angles can save time. Here are some tips:

• 0°, 30°, 45°, 60°, 90° : These are frequently used angles. Memorize their sine and cosine values.
• Use Patterns : Notice patterns in the values. For example, the sine of 30° is always half the sine of 60°.
• Practice : Regularly practice and use trigonometric functions to reinforce memorization.

Special Triangles

Special right triangles have angles that result in simple, easily memorable ratios. The two main types are the 30-60-90 triangle and the 45-45-90 triangle.

30-60-90 Triangle : In this triangle, the angles are 30°, 60°, and 90°. The sides are in the ratio 1: √3: 2. 

To find it, draw a right triangle with two additional angles that are roughly 30 and 60 degrees each to create a 30-60-90 triangle. You already know the ratios of sides. The smallest angle (30 degrees) is opposite the smallest side (QK). The hypotenuse, which has the largest side (PK), is located in opposition to the largest angle (90 degrees). The remaining 60-degree angle is opposite the square root of 3 (PQ).

45-45-90 Triangle : This triangle has angles of 45°, 45°, and 90°. The sides are in the ratio 1: 1: √2.

Draw a right triangle with two equal angles in the 45-45-90 triangle. The length of the hypotenuse is the square root of two; the remaining two sides are one. In order to calculate the cosine of 60 degrees, one would draw the triangle 30-60-90 and observe that the hypotenuse is two and the adjacent side is one. Therefore, 1/2 represents the cosine of 60 degrees.

Relevant Read : Choose The Right Math Tutor For Your Child

Importance of Understanding Trigonometric Functions Without a Calculator

A calculator has solved most of our problems but that does not mean that manual calculations have no benefits. You cannot always rely on using a calculator for everyday calculations. Sometimes, you have to do quick maths in certain situations where no calculator is available. 

. Let’s have a look at some of them.

• Manual trigonometry is a tool that promotes critical thinking . By understanding the ideas, people are able to see issues and come up with creative solutions rather than depending just on a calculator.
• Beyond using a calculator, trigonometry enhances problem-solving skills . It encourages people to break down complex problems, formulate strategies, and apply mathematical principles. 
• These manual calculations help in advanced studies and have diverse applications in various fields.

Where is Trigonometry used?

Trigonometry is used in various fields and by different experts. Some of them are given below:

• Navigation and Astronomy
• Surveying and Construction
• Engineering and Physics
• Computer Graphics
• Architecture

Interesting Information: If you want to apply a trigonometric function in a real-life scenario then take an example of a tree. Can you calculate the height of a tree without having to measure it from any measuring device?

If you know the distance between you and the tree and the angle where you are looking at the tree from the ground, then you can easily calculate the height of the tree.

Here’s how you can do it:

You know the formula of the tan function above. 

Tan (θ)= HeightDistance between you and the tree

Distance between you and the tree= HeightTan (θ)

Let’s assume the distance is 50 m and the angle is 45 degrees. The Height of the tree will be:

Height = Distance between you and the tree x Tan (θ)

Height = 50 x tan 45

Height = 50 m (since tan 45 = 1)

Choose a Tutor That Sharpens Your Skills

When practising trigonometry, the right tutor can make all the difference.

Meet My Tutor Source (MTS), a private tutoring company dedicated to sharpening your trigonometric skills. We provide one-on-one private tutoring sessions with experienced tutors all over the world. The flexibility of schedules and online options provided by MTS makes learning convenient, fitting seamlessly into your busy life.

So, if you want to sharpen your skills and be a manual trigonometry wizard, choose a math tutor from MTS. Book your lesson today with an expert.

Final Words

We've covered the basics of dealing with the sides and angles of triangles, explored useful tricks like using tables for quick solutions, and highlighted the importance of understanding trigonometry beyond just numbers.

The key takeaway is not about making math fancier; it's about making it more useful in everyday situations. By doing manual trigonometry, you're not just cramming numbers – you're sharpening your brain for problem-solving in fields like navigation, construction, computer graphics, and more. 

So, the next time you opt for manual calculations, remember it's a straightforward way to boost your practical math skills . For this, you can always take a free private class with an expert!

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Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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Man explains why we learn complex math problems that we'll never use in our lifetime

Usually, when asked about one subject that is the most complicated or least favorite, many people would instantly answer, "Mathematics." It is a common argument among many students that how learning math is pointless because they never use it in their future jobs. To some extent, it might be true. Unless our job demands us to solve mathematical problems, we never even have to think about calculus, algebra, trigonometry and other concepts we learned in school. However, this math enthusiast Hamza Alsamraee, a Stanford alumnus who majored in the subject, provided an answer for this argument through his LinkedIn post .

Being a math major, Alsamraee  agreed with  many who say they would never be solving elaborate problems at work. "I'll never solve a partial differential equation for work. But why is that the standard we judge the value of learning by?" he wrote. To this Stanford alumnus, saying that the college math we learned was "inapplicable in the real world" seemed silly. "College isn't a vocational training program. It's meant to provoke your curiosity, to expand your horizons," pointed out Alsamraee. Also, for those who think that college should only teach things that are accurately applicable to real-world jobs, he wrote, "If you want 'real world' learning, go to a boot camp/trade school, etc. There's nothing wrong with that. But trying to turn college into trade school is stupid ."

Alsamraee, the founder of a tech company called NewForm , also emphasized that the "useless" math concepts are the ones that  help us learn how to think. "How to analyze problems. How to weigh different perspectives. It's not a surprise the smartest people I've met in business also happen to have a lot of 'useless' knowledge," he wrote. As per the math enthusiast, "Curiosity shouldn't be constrained by capitalist pursuit." He mentioned that if people let capitalism control what we learn, we would end up being a "noninteresting person with little ability to think laterally" and that an AI agent will be outperforming us soon. "Go read about philosophy, math, history, whatever intrigues you. Who cares if it makes you money? It'll make you human," he concluded.

Men over forty share 10 valuable tips for men in their 30s and it's good advice

His post went viral, garnering over 15K on the platform. Later, Maheshwari Peri ( @maheshperi) shared it on X (previously Twitter), where it gained over 172K views after a student asked them why they had to learn trigonometry as it doesn't have real-life applicability. Not long after, people started sharing their views. "The response makes sense. Learning new unknown things opens up avenues and triggers curiosity. However, Trigonometry has real-life uses too. Without knowing Pythagoras's theorem, one might always take longer routes while they can reach their destination diagonally," commented @maddyb65 .

"It's laughable when someone says Trignometry and PDE have no real-world applicability. Just because students are not taught 'direct application' in class, does not mean there is no application in the world," wrote  @shivakjolad . "He's articulated his argument so well. Now we need systems of education and evaluation that force us to think and question," commented  @sammoorthy21 .

Speaking of college-level math, a Harvard math entrance exam from 1869 has been making its rounds on the internet. A Canadian mathematician and professor, Anthony Bonato ( @Anthony_Bonato ), shared those math questions pointing out that the calculator was not allowed back then. This left the internet flabbergasted, with many feeling lucky they now have access to calculators during math exams.

People share 10 common things from 50 years ago that only rich people can afford in today's time

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Use Active Listening to Help a Colleague Make a Hard Decision

• Cheryl Strauss Einhorn

Don’t jump straight to problem-solving.

Imagine a colleague is faced with a high-stakes decision. They’re likely stressed, conflicted, and overwhelmed. In these situations, many of us default to the role of problem-solver. We try to support our colleague by providing our opinion or offering a solution. But to effectively support decision makers in your organization, you need to step back from your own ego and just listen . This article outlines practical strategies for exercising four types of active listening: emotional, informational, analytical, and reflective. Active listening can be hard to do, but it’s a great skill to practice. It allows you to strengthen key relationships while giving decision makers the space to make decisions for themselves.

Arnaldo was the chief operating officer at a successful investment firm. Recently, the firm’s results had been underperforming expectations. This poor performance was due to one large investment that the chief investment officer, Russ, was committed to holding. Arnaldo had fielded several calls from investors who wanted Russ to sell the money-losing investment. So, when Russ asked for a meeting to discuss the fund’s performance, Arnaldo’s instinct was to make a pitch to sell — to solve the problem.

• Cheryl Strauss Einhorn is the founder and CEO of Decisive, a decision sciences company using her AREA Method decision-making system for individuals, companies, and nonprofits looking to solve complex problems. Decisive offers digital tools and in-person training, workshops, coaching and consulting. Cheryl is a long-time educator teaching at Columbia Business School and Cornell and has won several journalism awards for her investigative news stories. She’s authored two books on complex problem solving, Problem Solved for personal and professional decisions, and Investing In Financial Research about business, financial, and investment decisions. Her new book, Problem Solver, is about the psychology of personal decision-making and Problem Solver Profiles. For more information please watch Cheryl’s TED talk and visit areamethod.com .

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• Rational Number
• Powers of i
• Complex Form
• Partial Fractions
• Is Polynomial
• Leading Coefficient
• Leading Term
• Standard Form
• Complete the Square
• Synthetic Division
• Linear Factors
• Rationalize Denominator
• Rationalize Numerator
• Identify Type
• Convergence
• Interval Notation
• Pi (Product) Notation
• Boolean Algebra
• Truth Table
• Mutual Exclusive
• Cardinality
• Caretesian Product
• Age Problems
• Distance Problems
• Cost Problems
• Investment Problems
• Number Problems
• Percent Problems
• Addition/Subtraction
• Multiplication/Division
• Dice Problems
• Coin Problems
• Card Problems
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Most Used Actions

Number line.

• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

word-problems-calculator

• Middle School Math Solutions – Simultaneous Equations Calculator Solving simultaneous equations is one small algebra step further on from simple equations. Symbolab math solutions...

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