# Precalculus : Trigonometric Functions

## Example Questions

### Example Question #2 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

Solve for all x on the interval

Explanation:

Solve for all x on the interval

Remember Soh, Cah, Toa?

For this problem it helps to recall that

Since our tangent is equal to 1 in this problem, we know that our opposite and adjacent sides must be the same (otherwise we wouldn't get "1" when we divided them)

Can you think of any angles in the first quadrant which yield equal x and y values?

If you guessed  you guessed right! Remember that your  angle in the unit circle will give you a  triangle, which will have equal height and base.

### Example Question #3 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

The above triangle is a right triangle. Find the value of  (in degrees).

Explanation:

One can setup the relationship

After taking the arccosine,

the arccosine cancels out the cosine leaving just the value of .

### Example Question #4 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

What is the value of  (in degrees)?

Explanation:

One can setup the relationship

.

After taking the arctangent,

the arctangent cancels out the tangent and we are left with the value of .

### Example Question #1 : Evaluating Trig Functions

Solve for :

or

or

or

or

or

or

Explanation:

If the sine of an angle, in this case  is , the angle must be or .

Then we need to solve for theta by dividing by 3:

### Example Question #6 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

Which of the following could be a value of ?

Explanation:

Which of the following could be a value of ?

To begin, it will be helpful to recall the following property of tangent:

This means that if  our sine and cosine must have equal absolute values, but with opposite signs.

The only place where we will have equal values for sine and cosine will be at the locations halfway between our quadrantal angles (axes). In other words, our answer will align with one of the  angles.

Additionally, because our sine and cosine must have opposite signs (one negative and one positive), we need to be in either quadrant 2 or quadrant 4. There is only answer from either of those two, so our answer must be .

### Example Question #1 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

Find  if  and it is located in Quadrant I.

Explanation:

Since we know the value of the trigonometric function and the triangle is located in Quadrant I, we can draw the triangle and get a sense of it. If the opposite side is 1 and the hypotenuse is 2, we know that we're dealing with a 30-60-90 special triangle. And since the opposite side of the angle is 1, we know that the angle is .

### Example Question #8 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

Given the equation , what is one possible value of  ?

Explanation:

Find 1 possible value of  Given the following:

Recall that

So if , then

Thinking back to our unit circle, recall that cosine corresponds to the x-value. Therefore, we must be in quadrants II or III.

So, which angles correspond to an x-value of -0.5? Well, they must be the angles closest to the y-axis, which are our  increment angles.

This means our angle must be either

or

It must be , because 240 is not an option.

Note that there are technically infinte solutions, because we are not given a specific interval. However, we only need to worry about one.

### Example Question #1 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

What is the sine of an angle if a point on the terminal side of the angle is ?

Explanation:

Given the point on the coordinate plane , the origin to this point can be computed by the Pythagorean Theorem.

The hypotenuse of the right triangle formed by the origin and the point is .

The length of the triangle is 1 unit, and the height of the triangle is 5.

Sine of an angle is opposite side divided by the hypotenuse.

Rationalize the denominator.

### Example Question #2 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

Find the secant value of  if it's a point on the terminal side of an angle in standard position.

Explanation:

First, use the Pythagorean Theorem to solve for all the sides of the triangle. You know that the adjacent side is 4 units long, and the opposite side is -9 units long.

Using the Pythagorean Theorem, you should get a hypotenuse of .

Secant is defined as hypotenuse/opposite.

Thus, giving you an answer of .

### Example Question #3 : Find The Value Of The Sine Or Cosine Functions Of An Angle Given A Point On Its Terminal Side

Find the sine value of  if it is a point on the terminal side of an angle in standard position.