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This lesson will focus on analyzing and solving different trigonometric equations. For that purpose, inverse trigonometric functions and trigonometric identities will be used.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Riding a Ferris Wheel

On Sunday, Magdalena and her younger sister Paulina went to the amusement park Adventurally with their father. They all had a lot of fun going on numerous rides, including a Ferris Wheel. When they first saw it close up, the girls were so amazed by its size that they asked one of the workers for more details about it. The worker replied that it has a diameter of meters and it turns at a rate of revolutions per minute. When Magdalena's and Paulina's father heard this, he said that the height of their seat above the ground in meters after minutes can be modeled by the following function.
Then he asked his daughters two questions.
a How long will they have to wait until their seat is meters above the ground for the first time? Round the answer to the first decimal if needed.
b What height will the girls be at after minutes of riding the Ferris wheel? Round the answer to one decimal if needed.

## Presenting Inverse Trigonometric Functions

There are functions that undo the trigonometric functions, so to speak. These functions are called inverse trigonometric functions.

## Inverse Trigonometric Functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions. For example, the inverse sine is the inverse function of the sine function. The main inverse trigonometric functions are shown in the table below.

Trigonometric Function Inverse Trigonometric Function
The inverse trigonometric functions relate an input, which represents the ratio of two sides of a right triangle, to the measure of one of its two acute angles. The output angles are measured in radians. The domain of the corresponding trigonometric function must be restricted in order for its inverse to be defined as a function. The properties of the main inverse trigonometric functions are summarized in the following table.
Inverse Trigonometric Function Domain Range
All real numbers
Note that the inverse trigonometric functions are also called and respectively.
The inverse trigonometric functions are especially useful when solving equations involving trigonometric functions.

## Trigonometric Equations and Their Solutions

Contrary to trigonometric identities — which are true for all values of the variable for which both sides are defined — some equations involving trigonometric functions are true only for certain values of the variable. Now such equations will be presented.

## Trigonometric Equation

A trigonometric equation is an equation that includes one or more trigonometric functions. Consider the following example trigonometric equations.
If a trigonometric equation consists only trigonometric functions and constants, the solution is found by looking for the values of the argument that make the equation true. Solving trigonometric equations is similar to solving algebraic equations.
If possible, it is useful to rewrite a trigonometric equation such that they have the same trigonometric function on its both sides.
However, in cases where it is not possible, the inverse function to the trigonometric function that appears in the equation can be used.
Since the functions on the left-hand side are inverses, they cancel each other out and can be simplified to The value of the right-hand side, on the other hand, can be found by using a calculator.
Note that because trigonometric functions are periodic, they can have numerous angles corresponding to the same trigonometric value. This point can be illustrated by the following graph that shows at least four different solutions of the given equation. To sum up, the following facts can be used to solve trigonometric equations. Note that and written below are integers.

Equation Solutions
It can be concluded that trigonometric identities are a special case of trigonometric equation. Next, it will be shown how to solve different trigonometric equations.

## Decoding the Message with the Solutions to Trigonometric Equations

Today Magdalena’s math class started learning how to solve trigonometric equations. To encourage students to practice solving some more equations at home, Magdalena's teacher gave the students a fun task. A secret message has been encoded. The only way to read the message is to solve two equations to decode it. Input the answers to the first equation first. The smaller number in each pair should be written first. Write the numbers one after the other without any commas or spaces. Students who decode the message will get a bonus point on the upcoming test. The teacher said that
a
b

### Hint

a Gather all the trigonometric functions and constants on different sides of the equation and simplify.
b Use the Double-Angle Identity for cosine to write in terms of Then gather all the terms on one side of the equation and rewrite them as a product of two expressions.

### Solution

a To solve the given equation, start by rewriting it so that all the trigonometric function appear on one side of the equation and the constants on the other side.
Now, try to find the angle for which the value of equals The table of the trigonometric values of notable angles can be useful here.
It can be concluded that this equation has two possible solutions.
b Start by analyzing the given equation.
There are two different trigonometric functions involved in the equation, sine and cosine. Therefore, the first step is to rewrite one side of the equation in terms of the other. To do this, the Double-Angle Identity for cosine can be used.

Next, to solve this equation, rewrite the left-hand side as a product of two expressions.
Factor
By the Zero Product Property, in order for the equation to be true, at least one of the parentheses should be equal to
Note that adding to both sides of the Equation results in However, this is not possible, as the values of sine are always between and Therefore, this equation has no solution. Now consider Equation To find the value of or arcsine of think about the angles for which the value of sine is According to the table of the trigonometric ratios of common angles, there are two such angles in the interval between and Therefore, the given equation has two possible solutions.
Now that all the solutions have been found, the code mentioned at the beginning can be determined. In Part A, the solutions and were found, while the solutions from Part B were and The code can be found by combining these numbers in the order directed at the start.
Try it out to see the hidden message!

## Math Challenge With a Gift

Magdalena really liked the math joke that her teacher encoded with the solutions to the exercise. She and her friend Davontay decided to give each other two more equations to solve. If they solve them correctly, they will gift each other an envelope with another math joke they came up with or heard somewhere. Magdalena received the following two trigonometric equations from Davontay. He told her to write all the possible solutions in the form of an equation where is an integer number.
a
b

### Hint

a Use the Double-Angle Identity for cosine to rewrite in terms of
b Apply one of the Pythagorean Identities and simplify the equation.

### Solution

a In the first equation, the same trigonometric function appears two times but it has different arguments. Therefore, use the Double-Angle Identity for cosine to rewrite the equation.
Apply this identity and then simplify the equation.

In order to find the values of that satisfy this equation, think about the angles for which the cosine equals According to the table of the trigonometric ratios of common angles, there is at least one such angle.
Since cosine is a periodic function with a period of , for every integer the following angles are the solutions to the equation.
b Start by analyzing the given equation.
There are two trigonometric functions in the equation, tangent and secant. Recall that one of the Pythagorean Identities relates these two functions.
Substitute for into the equation and simplify it. Next, look for the angles for which tangent is equal to or
To find the general equation for the solutions, note that all these angles can be expressed in the following way.
Therefore, all the solutions to the equation can be given by the following equation where is an integer number.
After correctly solving both equations, Magdalena can finally read the joke that Davontay found for her. ## Making a Bet on Equations

The teacher gave the class a couple of exercises to solve for homework. She also warned that one of the equations has extraneous solutions and that the students should identify them. To make things interesting, Magdalena and Davontay decided to make a bet about which equation has extraneous solutions. After each chose an equation, they started solving them to see who guessed correctly. The winner will get the last piece of cake left in the fridge. To solve the equations, they must write all the solutions in radians such that

a
b
Who guessed correctly?

### Hint

a Use the Pythagorean Identity to rewrite in terms of Then factor the equation and apply the Zero Product Property.
b Move to the right side of the equation and then square both sides. Check the solutions by substituting them into the original equation and seeing if it remains true.

### Solution

a Start by examining the first equation.
Since there are two different trigonometric functions, one should be rewritten in terms of the other. To do so, use the Pythagorean Identity, but first rewrite the equation so that is isolated on one side of the equation.
Substitute for and simplify the equation.
Next, factor the expression on the left-hand side into two parentheses.
By the Zero Product Property, at least one of the parentheses must be equal to Two equations can be formed by applying this property. To solve these two equations, try to find the angles for which the values of sine are and For the first equation, there are two such angles — and — while for the second there is only one —
Finally, the solutions should be checked whether there are any extraneous solutions. To do that, substitute each of them into the original equation and see if it is true.
Simplify left-hand side

The other two solutions can be checked in a similar fashion.
Solution Substitute Evaluate True or False

Therefore, there are three solutions to the equation and none of them are extraneous.

b Again, begin by analyzing the given equation.
First, move to the right-hand side of the equation. Then, square each side of the equation.
Now, to rewrite one trigonometric function in terms of the other, use the Pythagorean Identity. Similarly to Part A, substitute for To find the solutions of this equation, look for the angles for which the sine equals The table with the trigonometric ratios of notable angles can be useful here.
Finally, substitute each of these solutions into the original equation in order to determine whether any of them is extraneous.
Solution Substitute Evaluate True or False