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 #64 : Unit Circle And Radians

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 #2 : 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 #1 : 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 #3 : 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 #72 : Unit Circle And Radians

For which values of , where  in the unit circle, is  undefined?

Explanation:

Recall that . Since the ratio of any two real numbers is undefined when the denominator is equal to  must be undefined for those values of  where . Restricting our attention to those values of  between  and  when  or . Hence,  is undefined when  or .

Example Question #1 : 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 #2 : 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 #2 : 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 .

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