# Trigonometry : Angles in the Unit Circle

## Example Questions

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### Example Question #1 : Angles In The Unit Circle

What are the ways to write 360o and 720o in radians?

Explanation:

on the unit circle.

on the unit circle.

### Example Question #1 : Angles In The Unit Circle

Which of the following is not an angle in the unit circle?

Explanation:

The unit circle is in two increments: , etc. and , etc. The only answer choice that is not a multiple of either  or  is .

### Example Question #1 : Angles In The Unit Circle

What is the equivalent of , in radians?

Explanation:

To convert from degrees to radians, use the equality

or .

From here we use  as a unit multiplier to convert our degrees into radians.

### Example Question #1 : Angles In The Unit Circle

What is the value of

Explanation:

To help with this one, draw a 45-45-90 triangle. With legs equal to 1 and a hypotenuse of .

Then, use the definition of tangent as opposite over adjacent to find the value.

Since the legs are congruent, we get that the ratio is 1.

### Example Question #2 : Angles In The Unit Circle

What is ?

Explanation:

Recall that on the unit circle, sine represents the y coordinate of the unit circle.

Then, since we are at 90˚, we are at the positive y axis, the point (0,1).

At this point on the unit circle, the y value is 1.

Thus  .

### Example Question #6 : Angles In The Unit Circle

Which of the following is NOT a special angle on the unit circle?

Explanation:

For an angle to be considered a special angle, the angle must be able to produce a  or a  triangle.

The only angle that is not capable of the special angles formation is .

### Example Question #7 : Angles In The Unit Circle

If  and , then =

The solution does not lie in the given interval.

Explanation:

We first make the substitution .

In the interval , the equation  has the solution .

Solving for ,

.

### Example Question #1 : Angles In The Unit Circle

What is the value of  from the unit circle?

Explanation:

From the unit circle, the value of

.

This can be found using the coordinate pair associated with the angle  which is .

Recall that the  pair are .

### Example Question #1 : Angles In The Unit Circle

What is

,

using the unit circle?

Explanation:

Recall that the unit circle can be broken down into four quadrants. Each quadrant has similar coordinate pairs basic on the angle. The only difference between the actual coordinate pairs is the sign on them. In quadrant I all signs are positive. In quadrant II only sine and cosecant are positive. In quadrant III tangent and cotangent are positive and quadrant IV only cosine and secant are positive.

has the reference angle of  and lies in quadrant IV therefore .

From the unit circle, the coordinate point of

corresponds with the angle .

### Example Question #4 : Angles In The Unit Circle

Give the exact value.

Use the unit circle to find:

None of the above

Explanation:

Locate  on the unit circle.

Sine is related to the why y value of the coordinate point because it is opposite/Hyp.

In other words the  pair of the point located on the unit circle that extends from the origin is .

The coordinate pair for  is  thus,

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