# Solving Trigonometric Equations

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.

#### Create learning materials about Solving Trigonometric Equations with our free learning app!

• Flashcards, notes, mock-exams and more
• Everything you need to ace your exams

## 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.

Illustration of a trigonometric cast diagram, Nicole Moyo - StudySmarter Originals

• 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.

$4\mathrm{sin}x°+3=00\le x\le 360°$

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

$4\mathrm{sin}x°+3=0\phantom{\rule{0ex}{0ex}}\mathrm{sin}x°=-\frac{3}{4}$

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.

$\mathrm{sin}x°=-\frac{3}{4}\phantom{\rule{0ex}{0ex}}x°={\mathrm{sin}}^{-1}\left(-\frac{3}{4}\right)\phantom{\rule{0ex}{0ex}}x°=-48.59°$

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.

$3rdquadratnt:x°=180°+48.59°=228.59°\phantom{\rule{0ex}{0ex}}4thquadrant:x°=360°-48.59°=311.41°$

## 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.

Unit circle. Image: Jim Belk, Public domain

## How do we solve quadratic trigonometric equations?

Quadratic trigonometric equations are second degree trigonometric equations. They can be solved by using the quadratic formula:$x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$

$2{\mathrm{sin}}^{2}a+3\mathrm{sin}a-1=0$

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

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

$2{x}^{2}+3x-1=0$

$a=2b=3c=-1\phantom{\rule{0ex}{0ex}}x=\frac{-\left(3\right)±\sqrt{{3}^{2}-4\left(2\right)\left(-1\right)}}{2\left(2\right)}\phantom{\rule{0ex}{0ex}}=\frac{-3±\sqrt{17}}{4}\phantom{\rule{0ex}{0ex}}$

Step 3: Replace your variable back as the function and take the inverse of the function to solve for the + equation. ( $±,meansthereare2solutions$)

${\mathrm{sin}}^{-1}\left(\frac{-3+\sqrt{17}}{4}\right)=18.11°\phantom{\rule{0ex}{0ex}}x=\mathrm{sin}\left(18.11\right)=0.28$

Step 4: Use the unit circle to determine the solution to the - equation as the domain of the inverse function is $\left[-1,1\right]$.

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

$x=\mathrm{\pi }-0.28\phantom{\rule{0ex}{0ex}}=2.86$

## 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:

${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{cos}x\mathrm{cos}y+\mathrm{sin}x\mathrm{sin}y=\mathrm{cos}\left(x-y\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{tan}}^{2}x+1=se{c}^{2}x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{cos}2x={\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x=2{\mathrm{cos}}^{2}x-1=1-2{\mathrm{sin}}^{2}x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{sin}2x=2\mathrm{sin}x\mathrm{cos}x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}$

$\mathrm{cos}x\mathrm{cos}\left(2x\right)+\mathrm{sin}x\mathrm{sin}\left(2x\right)=\frac{\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}$

Step 1: Simplify your equation with a known identity.

In this example, it is the difference formula for cosine: $\mathrm{cos}a\mathrm{cos}b+\mathrm{sin}a\mathrm{sin}b=\mathrm{cos}\left(a-b\right)$

$\mathrm{cos}x\mathrm{cos}\left(2x\right)+\mathrm{sin}x\mathrm{sin}\left(2x\right)=\frac{\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}c\mathrm{os}\left(x-2x\right)=\frac{\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{cos}\left(-x\right)=\frac{\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{cos}\left(x\right)=\frac{\sqrt{3}}{2},inthispartweusedthenegativeangletrigidentity:\mathrm{cos}\left(-x\right)=\mathrm{cos}\left(x\right)$

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, $x=\frac{\mathrm{\pi }}{6}andx=\frac{11\mathrm{\pi }}{6}$

## 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: $\mathrm{sin}2x,\mathrm{tan}\frac{x}{2},\mathrm{cos}3x,etc$The variables all have coefficients.

$\mathrm{cos}\left(2x\right)=\frac{1}{2}on\left[0,2\mathrm{\pi }\right)$

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

${\mathrm{cos}}^{_1}\left(\frac{1}{2}\right)=60°\phantom{\rule{0ex}{0ex}}\therefore possibleanglesare2x=\frac{\mathrm{\pi }}{3}and2x=\frac{5\mathrm{\pi }}{3}$

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

$2x=\frac{\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}x=\frac{\mathrm{\pi }}{6}\phantom{\rule{0ex}{0ex}}2x=\frac{5\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}x=\frac{5\mathrm{\pi }}{6}$

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

$Firstquadrant2x=\frac{\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}\mathrm{First}\mathrm{rotation}:2x=\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }\phantom{\rule{0ex}{0ex}}=\frac{7\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}\mathrm{x}=\frac{7\mathrm{\pi }}{6}\phantom{\rule{0ex}{0ex}}\mathrm{This}\mathrm{value}\mathrm{is}\mathrm{between}0\mathrm{and}2\mathrm{\pi }\mathrm{and}\mathrm{is}\mathrm{therefore}\mathrm{a}\mathrm{solution}\phantom{\rule{0ex}{0ex}}\mathrm{Second}\mathrm{Rotation}:2\mathrm{x}=\frac{\mathrm{\pi }}{3}+4\mathrm{\pi }\phantom{\rule{0ex}{0ex}}=\frac{13\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}\mathrm{x}=\frac{13\mathrm{\pi }}{6}\phantom{\rule{0ex}{0ex}}\mathrm{This}\mathrm{value}\mathrm{is}\mathrm{greater}\mathrm{than}2\mathrm{\pi }\mathrm{and}\mathrm{is}\mathrm{therefore}\mathrm{not}\mathrm{a}\mathrm{solution}.\phantom{\rule{0ex}{0ex}}\mathrm{Fourth}quadrant:2x=\frac{5\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}FirstRotation:2x=\frac{5\mathrm{\pi }}{3}+2\mathrm{\pi }\phantom{\rule{0ex}{0ex}}=\frac{11\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}\mathrm{x}=\frac{11\mathrm{\pi }}{6}\phantom{\rule{0ex}{0ex}}\mathrm{This}\mathrm{value}\mathrm{is}\mathrm{between}\mathrm{the}\mathrm{range}\mathrm{and}\mathrm{is}\mathrm{therefore}\mathrm{a}\mathrm{solution}.\phantom{\rule{0ex}{0ex}}\mathrm{Second}\mathrm{Rotation}:2\mathrm{x}=\frac{5\mathrm{\pi }}{3}+4\mathrm{\pi }\phantom{\rule{0ex}{0ex}}2\mathrm{x}=\frac{17\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}\mathrm{x}=\frac{17\mathrm{\pi }}{6}\phantom{\rule{0ex}{0ex}}\mathrm{This}\mathrm{value}\mathrm{is}\mathrm{not}\mathrm{within}\mathrm{your}\mathrm{range}\mathrm{and}\mathrm{thefore}\mathrm{cant}\mathrm{be}\mathrm{a}\mathrm{solution}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

## 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.
• A unit circle is a circle that has a radius of 1 and is used to illustrate particular common angles.
• Quadratic trig equations can be solved with the quadratic formula:$x=\frac{-b±±\sqrt{{b}^{2}-4ac}}{2a}$
• 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.

#### Flashcards in Solving Trigonometric Equations 3

###### Learn with 3 Solving Trigonometric Equations flashcards in the free StudySmarter app

We have 14,000 flashcards about Dynamic Landscapes.

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.

StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

##### StudySmarter Editorial Team

Team Math Teachers

• Checked by StudySmarter Editorial Team