# Precalculus : Trigonometric Functions

## Example Questions

### 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 #3 : 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 #5 : 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

### Example Question #11 : Find The Value Of Any Of The Six Trigonometric Functions

Choose the answer which is equivalent to the following trig expression:

Explanation:

Choose the answer which is equivalent to the following trig expression:

Begin by finding the location of our given angle. If we start at 0 on the unit circle and go clockwise, every quadrant covers

Therefore, our given angle will correspond to , which is the bottom half of the y-axis.

Now, because cosine corresponds to the x-value, we know that this expression must be equivalent to 0. If we are on the y-axis, we have no x-value, and therefore, cosine must equal 0

### Example Question #11 : Trigonometric Functions

Which of the following is equivalent to the given expression?

Explanation:

Which of the following is equivalent to the given expression?

To simplify cotangent expressions, we can think of the expression as tangent and then simply take the reciprocal. So:

, which is undefined.

So,

### Example Question #13 : Find The Value Of Any Of The Six Trigonometric Functions

Simplify the following expression:

Explanation:

Simplify the following expression:

I would begin here by recalling that secant is the reciprocal of cosine. Therefore, we can take the cosine of the given angle and then find its reciprocal.

So,

(Because cosine refers to x-values and  lies on the x-axis)

Therefore,

Because .

### Example Question #14 : Find The Value Of Any Of The Six Trigonometric Functions

Calculate the value of the following trig function:

Explanation:

Calculate the value of the following trig function:

This is a problem which can be instantly solved by a calculator, provided it isn't in radian mode.

However, we are going to run through how to figure this out without a calculator, because knowing how to do it is far more powerful.

Begin by placing the angle in the unit circle.   is a multiple of , more specifically, it is three times ninety.

Because we know our angle is three times ninety, we know the angle we are dealing with is one of our four "quadrantal" angles. These are the four that make up our x-y grid.

So,  is the angle directly opposite of . This means our x-value must be zero, and our y-value must be .

Now, since we know that sine is basically our y-value, the value of  must be equal to .

### Example Question #15 : Find The Value Of Any Of The Six Trigonometric Functions

In a 3-4-5 right triangle, which of the following is a possible value for ?