# Precalculus : Find the value of any of the six trigonometric functions

## Example Questions

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### Example Question #1 : Trigonometric Functions

Solve the following:

Explanation:

Rewrite  in terms of sine and cosine functions.

Since these angles are special angles from the unit circle, the values of each term can be determined from the x and y coordinate points at the specified angle.

Solve each term and simplify the expression.

### Example Question #1 : Find The Value Of Any Of The Six Trigonometric Functions

Find the value of .

Explanation:

Using trigonometric relationships, one can set up the equation

.

Solving for ,

Thus, the answer is found to be 29.

### Example Question #1 : Trigonometric Functions

Find the value of .

Explanation:

Using trigonometric relationships, one can set up the equation

.

Plugging in the values given in the picture we get the equation,

.

Solving for ,

.

Thus, the answer is found to be 106.

### Example Question #3 : Find The Value Of Any Of The Six Trigonometric Functions

Find all of the angles that satistfy the following equation:

OR

OR

Explanation:

The values of  that fit this equation would be:

and

because these angles are in QI and QII where sin is positive and where

.

is incorrect, because it includes inputs that provide negative values such as:

Thus the answer would be each  multiple of  and  , which would provide the following equations:

OR

### Example Question #1 : Find The Value Of Any Of The Six Trigonometric Functions

Evaluate:

Explanation:

To evaluate , break up each term into 3 parts and evaluate each term individually.

Simplify by combining the three terms.

### Example Question #1 : Find The Value Of Any Of The Six Trigonometric Functions

What is the value of  ?

Explanation:

Convert  in terms of sine and cosine.

Since theta is  radians, the value of  is the y-value of the point on the unit circle at  radians, and the value of  corresponds to the x-value at that angle.

The point on the unit circle at  radians is .

Therefore,  and .  Substitute these values and solve.

### Example Question #2 : Find The Value Of Any Of The Six Trigonometric Functions

Solve:

Explanation:

First, solve the value of .

On the unit circle, the coordinate at  radians is .  The sine value is the y-value, which is .  Substitute this value back into the original problem.

Rationalize the denominator.

### Example Question #1 : Find The Value Of Any Of The Six Trigonometric Functions

Explanation:

To evaluate , solve each term individually.

refers to the x-value of the coordinate at 60 degrees from the origin.  The x-value of this special angle is .

refers to the y-value of the coordinate at 30 degrees.  The y-value of this special angle is .

refers to the x-value of the coordinate at 30 degrees.  The x-value is .

Combine the terms to solve .

### Example Question #4 : Find The Value Of Any Of The Six Trigonometric Functions

Find the value of

.

Explanation:

The value of  refers to the y-value of the coordinate that is located in the fourth quadrant.

This angle  is also  from the origin.

Therefore, we are evaluating .

### Example Question #1 : Find The Value Of Any Of The Six Trigonometric Functions

Simplify the following expression:

Explanation:

Simplify the following expression:

Begin by locating the angle on the unit circle. -270 should lie on the same location as 90. We get there by starting at 0 and rotating clockwise

So, we know that

And since we know that sin refers to y-values, we know that

So therefore, our answer must be 1

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