Math 3 Name_______________________________
5-3 Unit Circle Exact Values in Degrees

I can use sine and cosine functions to describe rotations of circular functions

I can use degree measures to measure angles and rotations

I can evaluate exact values for sine, cosine, and tangent around the unit circle
Recall from geometry that an angle is the union of two rays (its sides ) with the same endpoints (its vertex ).
An angle can be thought of as being generated by rotating a ray either counterclockwise or clockwise around its endpoint from one position to another. For instance, in

AQB below, you can think of QB as the image of QA under a counterclockwise rotation with center Q . The measure of an angle is a number the represents the size and direction of rotation used to generate the angle. In trigonometry , angles generated by a counterclockwise rotation are measured with positive numbers, and angles generated by clockwise rotations are measured with negative numbers.
So far to this point in your illustrious mathematical careers, you have learned to measure angles in degrees . For instance, the measure of

AQB above has a measure of 50 degrees. If QA is considered the initial side and QB is its image under a counterclockwise rotation (called the terminal side ), then m

AQB = 50

. However, if QB was the initial side and rotated clockwise around Q so that QA is the terminal side, then m

AQB =

50

. Be sure you understand the idea of positive and negative angle measures before moving on!!!!!
Another unit for measuring rotations is the revolution. Revolution is related to degrees by the following formula: 1 revolution counterclockwise = 360

. Some examples for equivalent degree measures are shown below.
Fig. A Fig. B Fig. C
1
8
revolution counterclockwise
2
3
revolution clockwise 1
1
4
revolution counterclockwise
1

360

45 360
 
240
5
4

360

450
8 3
Notice that the same rotation can have many different magnitudes. Fill in the blanks below

2
In figure B, the measurement of the angle is revolution clockwise or -240 degrees. This is equivalent
3 to a _____ revolution counterclockwise and ______ degrees.
In figure C, the measurement of the angle is 1
1
4
revolutions counterclockwise or 450 degrees. This is equivalent to both a _______ revolution counterclockwise and ______ degrees (rotating counterclockwise) and ______ revolution clockwise and ______ degrees (rotating clockwise).
Practice
If the terminal side of an angle rotates
1
6 of the angle formed? of a revolution from the initial side, how is the degree measure
If an angle is 150 , what is the measure of the angle in revolutions?
If the terminal side of an angle rotates
3
4 of the angle formed? of a revolution from the initial side, how is the degree measure

3.
4.
Use the provided grid and a protractor to complete the following activity. The circle on the grid is called a unit circle because it has a radius of 1 unit.
1. Use a protractor to mark the rotation of the point (1, 0) under a rotation of 50°
(in math, counterclockwise is considered a positive). Call this new point P
1 counterclockwise
2. Use the grid to estimate the coordinate of the x-coordinate and the y-coordinate of P
1
and the slope of OP
1
. x-coordinate:_______ y-coordinate:_______ slope of OP
1
:_______
Set your calculator to degree mode. Use your calculator to find sin cos 50 = _______
Use a protractor to mark the rotation of the point (1, 0) under a rotation of 100 degrees. Call this new point P
2 sin 50 = _______
50° , cos 50° tan 50 = _______
, and tan 50°
5. Use the grid to estimate the coordinate of the x-coordinate and the y-coordinate of P
2
and the slope of OP
2 x-coordinate:_______ y-coordinate:_______ slope of OP
2
:_______
6.
7.
Set your calculator to degree mode. Use your calculator to find sin 100° , cos 100° , and tan 100° cos 100 = _______ sin 100 = _______ tan 100 = _______
NOTES:
Look back at your work from 1 – 3 and 4 – 6. What relations do you see between the coordinates of the image of (1, 0) under a rotation of magnitude and tan

?
 and the values of cos

, sin

,
Use the first quadrant and the fact that the radius of the unit circle is 1 to explain why your answer to number (7) is correct.

8. Without a calculator , find sin 90 ( hint: where is 90 located on the circle? What is the coordinate of that point?
) Find
 
.
9. How is tan

related to sin

and cos

?
For angles on the unit circle, generally represented by the Greek letter 𝜃 , it is assumed that the vertex of the angle is (0, 0) and the initial side of the angle has an endpoint of (1, 0)
10. Based on what you have discovered, what is the coordinate of a point on the unit circle that has an angle of 30° ? Check your answer using your protractor and the unit circle.
11. What angle has coordinates of (0.7071, 0.7071).
12. Find the coordinates (to the nearest thousandth) on the unit circle if 𝜃 is the following: a. 40° _______________ b. 150° _______________ c.
1
3
revolution ____________ d. −60° _______________
13. The coordinate (0.3907, 0.9205) is on the unit circle. Using your knowledge of the symmetry of a circle, what other coordinates must also be on the circle?
14. What angle represents a rotation from (1, 0) to the coordinate (0.3907, 0.9205)?
15. Using your knowledge of the symmetry of a circle (and NOT any trigonometric functions), find the angles that represent the other coordinates from number (13).

( except for #18 )
While we can use our sandbox trigonometry to find values on the unit circle, we will generally have to round those values. However there are some exact value coordinates on the unit circle, such as the coordinates at angle rotations of 0°. 90°, 180°, 270° , and 360° (and all their multiples).
16a. Without using a calculator, find the coordinates on the unit circle if 𝜃 is the following: a. 90° ________ b. 180° ________ c. 270° ________ d. 360° ________
16b. Based on your answers to part (a), find the following exact values : a.
  
_____ b.
  
_____ c.
  
_____ d.
  
_____
Besides the coordinates that are on the x-axis and y-axis, there are others that have exact values.
17. Look at problem (10). What is your y-coordinate? _________.
That means that 𝑠𝑖𝑛(30°) = ______.
Therefore, 𝑠𝑖𝑛(30°) has an exact value .
1
18. What is the decimal equivalent of
√2
? _________. Which previous problem on this worksheet
1 have you seen this value ? Note: many books will rationalize by multiplying by
2
“fancy one” 
1

2

2
2
2 2
That means that 𝑠𝑖𝑛( ) = ________ and 𝑐𝑜𝑠( ) = _______.
So 𝑠𝑖𝑛( ) and 𝑐𝑜𝑠( ) also have exact values.
So you now of course are wondering “what other values of 𝜃 have exact values? And where do these exact values come from?” Let’s find out.

Do the triangle ratios to the right look familiar?
19. Use the appropriate ratio to find sin (30°)
20. Use the appropriate ratio to find cos (30°)
21. Use the appropriate ratios to calculate
 
22. Use the appropriate ratios to calculate
 

23. On the unit circle to the right, draw a 150 rotation from
 
24. How can we use the drawing from problem (23) to find the
 
25. What is
26. What is
 

Quick Mental Math facts pertinent to the Unit Circle
180 – 30 =
180 – 45 =
180 – 60 =
180 + 30 =
180 + 45 =
180 + 60 =
360 – 60 =
360 – 45 =
360 – 30 =
After calculating the mental math facts, label the angles on the unit circle in the appropriate quadrant.

NOTES:


Find the following unit circle values. As much as possible, do NOT look at the unit circle on the previous page.
27. Find 𝑠𝑖𝑛(120°) 28. Find 𝑐𝑜𝑠(240°)
29. Find the below values. Use the circles to help you if necessary.








30. What are the possible values for tangent? List them below, with the appropriate calculations.
31. Find the following values WITHOUT looking at your unit circle on the previous page.


















