# Precalculus : Trigonometric Functions

## Example Questions

### Example Question #1 : Trigonometric Functions

Which of the following is the correct domain of , where  represents an integer?

Explanation:

The cotangent graph only has a period of  intervals and is most similar to the tangent graph.  The domain of cotangent exists everywhere except every  value since an asymptote exists at those values in the domain.

The y-intercept of 3 shifts the cotangent graph up by three units, so this does not affect the domain.

Therefore, the graph exists everywhere except , where  is an integer.

### Example Question #2 : Determine The Domain Of A Trigonometric Function

What is the domain of the following function?

Explanation:

All x values make the function work. Thus, making the domain   . They're parentheses instead of brackets because parentheses are used when you can't actually use the specific value next to it. It is impossible to use infinity which makes parentheses appropriate. Brackets are used when you CAN use the specific value next to it.

### Example Question #1 : Trigonometric Functions

What is the domain of the following function:

Explanation:

All x values work for the function. Thus, making the domain all real numbers. Parentheses are required because you can never actually use the number infinity.

### Example Question #4 : Determine The Domain Of A Trigonometric Function

What is the domain of .

Explanation:

If you look at a graph of the function, you can see that every curve has a vertical asymptote that repeats every  radians in the positive and negative x-direction, starting at  radians.  Also, the curve has a length that stretches  radians which makes the domain .

### Example Question #5 : Determine The Domain Of A Trigonometric Function

What is the restriction of the domain of the function given by:

For all the answer choices below,  is any integer.

Explanation:

has restrictions on its domain such that

, where  is any integer.

To determine the domain for

we equate the terms within the secant function and set them equal to the original domain restriction.

Solving for

The new domain restriction is:

where  is an integer

### Example Question #1 : Find The Value Of Any Of The Six 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 #2 : 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 #3 : Find The Value Of Any Of The Six 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 #1 : 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

Evaluate: