## Trig. Equations Examples using CAST Diagrams

These lessons, with videos, examples and step-by-step solutions help A Level Maths students learn to solve trigonometric problems.

Related Pages Trigonometric Functions Trigonometric Graphs Trigonometric Identities Lessons On Trigonometry More Lessons for A Level Maths

What is the CAST diagram? The Cast diagram helps us to remember the signs of the trigonometric functions in each of the quadrants. The CAST diagram is also called the Quadrant Rule or the ASTC diagram.

In the first quadrant, the values are all positive. In the second quadrant, only the values for sin are positive. In the third quadrant, only the values for tan are positive. In the fourth quadrant, only the values for cos are positive.

Note that the mnemonic CAST goes anticlockwise starting from the 4th quadrant.

The following diagram shows how the CAST diagram or Quadrant rule can be used. Scroll down the page for more examples and solutions.

Quadrant Rule or CAST diagram

Using the Quadrant Rule to solve trig. equations

Quadrant Rule for solving trig equations with different ranges

Core (2) Graphs of Trigonometric Functions (4) - CAST Diagram or Unit Circle

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A trigonometric equation is an equation that consists of a trigonometric function. These functions include sine, cosine, tangent, cotangent, secant and cosecant. Depending on the type of trigonometric equation, they can be solved using a CAST diagram, the quadratic formula, one of the various trigonometric identities available, or the unit circle.

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A trigonometric equation is an equation that consists of a trigonometric function. These functions include sine, cosine, tangent, cotangent, secant and cosecant. Depending on the type of trigonometric equation, they can be solved using a CAST diagram, the quadratic formula, one of the various trigonometric identities available, or the unit circle.

## How do we use a CAST diagram when solving trigonometric equations?

A CAST diagram is used to solve trigonometric equations. It helps us remember the signs of the trigonometric functions in each quadrant and what happens to the angle that needs to be calculated, depending on the trigonometric function used.

- All trig functions are positive in the first quadrant.
- Only sine is positive in the second quadrant.
- Only tangent is positive in the third quadrant.
- Only cosine is positive in the fourth quadrant.

When using the CAST diagram, you will first isolate the trig function, calculate your acute angle, and then use the diagram to solve for the solutions. You can use this method to solve linear trig equations, trig equations involving a single function, and use your calculator.

4 sin x ° + 3 = 0 0 ≤ x ≤ 360 °

Step 1: Rearrange the equation to have the trig function be on its own.

4 sin x ° + 3 = 0 sin x ° = - 3 4

Step 2 : Calculate the value of your acute angle using the inverse of your trig function. Note that the negative will always be ignored when calculating the acute angle.

sin x ° = - 3 4 x ° = sin - 1 ( - 3 4 ) x ° = - 48 . 59 °

Step 3: Based on the sign of the function, determine the quadrants of the solutions and use the information from this to solve the equation.

In our example, sine is negative. Therefore our solutions are in the 3rd (180 ° + x °) and 4th (360 ° -x °) quadrants.

3 r d q u a d r a t n t : x ° = 180 ° + 48 . 59 ° = 228 . 59 ° 4 t h q u a d r a n t : x ° = 360 ° - 48 . 59 ° = 311 . 41 °

## What is the unit circle in trigonometry?

- A unit circle is a circle that has a radius of 1 and is used to illustrate particular common angles.

## How do we solve quadratic trigonometric equations?

Quadratic trigonometric equations are second degree trigonometric equations. They can be solved by using the quadratic formula: x = - b ± b 2 - 4 a c 2 a

2 sin 2 a + 3 sin a - 1 = 0

Step 1: Replace your trig function with a variable of your choice.

In our example, we will say let sin (a) = x

2 x 2 + 3 x - 1 = 0

Step 2: Use the quadratic formula to solve for your variable.

a = 2 b = 3 c = - 1 x = - ( 3 ) ± 3 2 - 4 ( 2 ) ( - 1 ) 2 ( 2 ) = - 3 ± 17 4

Step 3: Replace your variable back as the function and take the inverse of the function to solve for the + equation. ( ± , m e a n s t h e r e a r e 2 s o l u t i o n s )

sin - 1 ( - 3 + 17 4 ) = 18 . 11 ° x = sin ( 18 . 11 ) = 0 . 28

Step 4: Use the unit circle to determine the solution to the - equation as the domain of the inverse function is - 1 , 1 .

Due to sine being positive in the first and second quadrants, the second solution would be:

x = π - 0 . 28 = 2 . 86

## How do we use identities to solve trigonometric equations?

Identities are used to solve trigonometric functions by simplifying the equation and then solving mainly by using the unit circle.

Here are a few important trigonometric formulas you should know:

sin 2 x + cos 2 x = 1 cos x cos y + sin x sin y = cos ( x - y ) tan 2 x + 1 = s e c 2 x cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x sin 2 x = 2 sin x cos x tan x = sin x cos x

cos x cos ( 2 x ) + sin x sin ( 2 x ) = 3 2

Step 1: Simplify your equation with a known identity.

In this example, it is the difference formula for cosine: cos a cos b + sin a sin b = cos ( a - b )

cos x cos ( 2 x ) + sin x sin ( 2 x ) = 3 2 c os ( x - 2 x ) = 3 2 cos ( - x ) = 3 2 cos ( x ) = 3 2 , i n t h i s p a r t w e u s e d t h e n e g a t i v e a n g l e t r i g i d e n t i t y : cos ( - x ) = cos ( x )

Step 2: Use the unit circle to determine the values of your angle (x).

In our example, we will focus on the 4th and 1st quadrants as cosine is positive in those quadrants.

Therefore, x = π 6 a n d x = 11 π 6

## How do we solve trigonometric equations with multiple angles?

Trigonometric equations with multiple angles are solved by first rewriting the equation as an inverse, determining which angles satisfy the equation and then dividing these angles by the number of angles. In solving these, you will most likely have more than two solutions as when you have a function in this form: cos (nx) = c, you will need to go around the circle n times.

Trigonometric equations with multiple angles look like this: sin 2 x , tan x 2 , cos 3 x , e t c The variables all have coefficients.

cos 2 x = 1 2 o n [ 0 , 2 π )

Step 1: Determine the quadrants of your initial solutions and the possible angles by using the unit circle.

cos _ 1 ( 1 2 ) = 60 ° ∴ p o s s i b l e a n g l e s a r e 2 x = π 3 a n d 2 x = 5 π 3

Step 2: Calculate the value of your initial solutions by dividing the possible angle by the number of angles.

2 x = π 3 x = π 6 2 x = 5 π 3 x = 5 π 6

Step 3: Determine your other solutions by revolving around the circle by the number of angles and only selecting the answers within your range.

F i r s t q u a d r a n t 2 x = π 3 First rotation : 2 x = π 3 + 2 π = 7 π 3 x = 7 π 6 This value is between 0 and 2 π and is therefore a solution Second Rotation : 2 x = π 3 + 4 π = 13 π 3 x = 13 π 6 This value is greater than 2 π and is therefore not a solution . Fourth q u a d r a n t : 2 x = 5 π 3 F i r s t R o t a t i o n : 2 x = 5 π 3 + 2 π = 11 π 3 x = 11 π 6 This value is between the range and is therefore a solution . Second Rotation : 2 x = 5 π 3 + 4 π 2 x = 17 π 3 x = 17 π 6 This value is not within your range and thefore cant be a solution .

## Solving Trigonometric Equations - Key takeaways

- When using the CAST diagram, you will first isolate the trig function, calculate your acute angle and then use the diagram to solve for the solutions.
- Quadratic trig equations can be solved with the quadratic formula: x = - b ± ± b 2 - 4 a c 2 a
- Identities are used to solve trigonometric functions by simplifying the equation and then solving using the unit circle.
- In solving trig functions with multiple angles, you will most likely have more than two solutions as when you have a function in this form: cos (nx) = c, you will need to go around the circle n times.

## Frequently Asked Questions about Solving Trigonometric Equations

--> how do we solve trigonometeric equations.

Trigonometric equations can be solved using a CAST diagram, the quadratic formula, trigonometric identities and the unit circle.

## --> How do we solve trigonomteric equations in radians?

Step 1: Determine the quadrants of your initial solutions and the possible angle, by using the unit circle.

Step 2: Calculate the value of your initial solutions by dividing the possible angle by the number of angles.

## --> How to solve trigonometric equations algebraically?

Step 1: Rearrange the equation to have the trig function on its own.

Step 2: Calculate the value of your acute angle, using the inverse of your trig function. Note that the negative will always be ignored when calculating the acute angle.

Step 3: Based on the sign of the function, determine the quadrants of the solutions and use the information from this to solve the equation.

## --> How to solve trigonometric equations using identities ?

Step 1: Simplify your equation with a known identity.

Step 2: Use the unit of circle to determine the values of your angle.

## --> How to solve for trigonometric equations for the given domain?

Step 1: Determine the quadrants of your initial solutions and the possible angles, by using the unit circle.

Step 2: Calculate the value of your initial solutions by dividing the possible angle by the number of angles.

Step 3: Determine your other solutions by revolving around the circle by the number of angles and only selecting the answers within your range.

## Final Solving Trigonometric Equations Quiz

Solving trigonometric equations quiz - teste dein wissen.

What is a trigonometric equation?

Show answer

A trigonometric equation is one that includes a trigonometric function which could be: sine, cosine, tangent, cotangent, secant and cosecant.

Show question

What is a CAST diagram?

A CAST diagram is one that helps us remember the signs of the trigonometric functions in each of the quadrants and what happens to the angle that needs to be calculated, depending on the trigonometric function used.

In which quadrants is tangent positive?

Tangent is positive in the first and third quadrants.

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## Trig Equations Revision

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Solving the basic trig equations is pretty easy, but what happens if we’ve got a function which has been stretched or translated ?

Make sure you are happy with the following topics before continuing.

## Trig Graphs

First, we need to find an initial solution.

So, for example, let’s say we want to find the values of x when \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} .

We want to draw the graph, and mark on a horizontal line where the condition is met, i.e. where \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} .

By inspection, we know that the initial solution is at 30° , and we can see it repeats at every 180° .

We’ll denote this 30° ± 180°n .

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## CAST Diagrams

The CAST diagram is a handy tool to show us where values of the standard trig functions are positive .

So, here’s the breakdown:

- For 0° < x < 90° , ALL of \textcolor{blue}{\sin x}, \textcolor{limegreen}{\cos x} and \textcolor{red}{\tan x} are positive
- For 90° < x < 180° , ONLY \textcolor{blue}{\sin x} is positive
- For 180° < x < 270° , ONLY \textcolor{red}{\tan x} is positive
- For 270° < x < 360° , ONLY \textcolor{limegreen}{\cos x} is positive

Think back to plotting the Unit Circle, in the Trig Basics section.

Let’s say we want to find the values of x such that \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} , as before.

We know that there is a solution when x = 30° .

First, we plot the point on the diagram, then find the corresponding angles:

From here, we can see that \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} when x = 30° ± 360°n or 210° ± 360°n , or, more concisely, 30° ± 180°n .

We ignored the two solutions where \textcolor{red}{\tan x} is not positive, i.e. 90° < x < 180° and 270° < x < 360° .

## Dealing With Trig Transformations

Transformations pose a little bit of a problem… See, CAST diagrams become much harder to navigate now. You’re much better off sketching out the function and solving using the horizontal line technique.

So, let’s just begin with an example.

We have f(x) = \cos 3x . Find the values of x \in \lbrack 0°, 360°\rbrack * such that f(x) = \textcolor{purple}{\dfrac{1}{2}} .

* This is just set notation, meaning 0° \leq x \leq 360°

Well, we have a series of solutions, but they’re not immediately obvious.

What we can do instead is plot the regular \textcolor{limegreen}{\cos x} graph on an interval three times as large as the proposed interval, and divide our solutions there by 3 .

\textcolor{limegreen}{\cos x} = \textcolor{purple}{\dfrac{1}{2}} has solutions 60°, 300°, 420°, 660°, 780°, 1020° .

Therefore, \cos 3x = \textcolor{purple}{\dfrac{1}{2}} has solutions 20°, 100°, 140°, 220°, 260°, 340° .

## Trig Equations Example Questions

Question 1: Sketch the \sin x graph and by inspection, find the solutions of \sin x = \dfrac{-\sqrt{3}}{2} , in the interval -360° \leq x \leq 360° .

x = \sin ^{-1} \left( \dfrac{\sqrt{3}}{2}\right) = -120°, -60°, 240°, 300°

Question 2: Using a CAST diagram, find the values of x such that \cos x = 0.2588 .

\cos x = 0.2588 gives x = 75° .

Question 3: For f(x) = 2\sin \dfrac{3x}{2} , find the values of x \in \lbrack -2\pi, 2\pi \rbrack where f(x) = 1 .

For f(x) = 1 , we want \sin \dfrac{3x}{2} = \dfrac{1}{2} .

If we look for values of -3\pi \leq x \leq 3\pi where \sin x = \dfrac{1}{2} , and multiply our values of x by a scale factor of \dfrac{2}{3} , we have our new set of solutions.

\sin x = \dfrac{1}{2} occurs at \dfrac{-11\pi}{6}, \dfrac{-7\pi}{6}, \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{13\pi}{6}, \dfrac{17\pi}{6}

Therefore the solution of f(x) = 1 is x = \dfrac{-11\pi}{9}, \dfrac{-7\pi}{9}, \dfrac{\pi}{9}, \dfrac{5\pi}{9}, \dfrac{13\pi}{9}, \dfrac{17\pi}{9} .

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## The CAST Diagram

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Think back to plotting the

The quadrant rule or CAST diagram is very useful to know as it saves ... Using the quadrant rule to solve simple trigonometric equations.

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