Download 5-3 Unit Circle Exact Values (Degrees)

Math 3
5-3 Unit Circle Exact Values in Degrees
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Name_______________________________
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
2
1
1 revolution counterclockwise
revolution counterclockwise
revolution clockwise
8
3
4
1
2
5
 360  45
  360  240
 360  450
8
3
4
Notice that the same rotation can have many different magnitudes. Fill in the blanks below
In figure B, the measurement of the angle is
2
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
revolutions counterclockwise or 450 degrees. This is
4
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
of a revolution from the initial side, how is the degree measure
6
of the angle formed?
If an angle is 150 , what is the measure of the angle in revolutions?
If the terminal side of an angle rotates
of the angle formed?
3
of a revolution from the initial side, how is the degree measure
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) along the circle under a rotation of 50°
counterclockwise (in math, counterclockwise is considered a positive). Call this new point P1
2.
Use the grid to estimate the x-coordinate and the y-coordinate of P1 and the slope of OP1
(estimate to two decimal places).
x-coordinate:_______
3.
y-coordinate:_______
slope of OP1 :_______
Set your calculator to degree mode. Use your calculator to find sin 50°, cos 50°, and tan 50°
cos 50 = _______
sin 50 = _______
tan 50 = _______
4.
Use a protractor to mark the rotation of the point (1, 0) under a rotation of 100 degrees. Call this
new point P2
5.
Use the grid to estimate the coordinate of the x-coordinate and the y-coordinate of P2 and the
slope of OP2
x-coordinate:_______
6.
slope of OP2 :_______
Set your calculator to degree mode. Use your calculator to find sin 100°, cos 100°, and tan 100°
cos 100 = _______
7.
y-coordinate:_______
sin 100 = _______
tan 100 = _______
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 the values of cos  ,sin  ,
and tan  ?
NOTES:
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 sin 180  .
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.
c.
40°
1
3
_______________
revolution
____________
b.
150° _______________
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).
NO CALCULATOR FROM THIS POINT ON!!! (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.
16b.
Without using a calculator, find the coordinates on the unit circle if 𝜃 is the following:
a.
90° ________
b.
180° ________
c.
270° ________
d.
360° ________
Based on your answers to part (a), find the following exact values:
a.
cos  90  _____
b.
tan 180  _____
c.
sin  270  _____
d.
cos  360  _____
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.
18.
What is the decimal equivalent of
1
√2
? _________. Which previous problem on this worksheet
have you seen this value? Note: many books will rationalize
1
by multiplying by
2
“fancy one” 
1

2
That means that 𝑠𝑖𝑛(
So 𝑠𝑖𝑛(
) and 𝑐𝑜𝑠(
2
2

2
2
) = ________ 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 sin  45 and cos  45 .
22.
Use the appropriate ratios to calculate sin  60 and cos  60 .
23.
On the unit circle to the right, draw a 150
rotation from 1,0 
24.
How can we use the drawing from problem (23)
to find the sin 150 ?
25.
What is cos 150  ? tan 150 ?
26.
What is sin  240  ?
Quick Mental Math facts pertinent to the Unit Circle
180 – 30 =
180 + 30 =
360 – 30 =
180 – 45 =
180 + 45 =
360 – 45 =
180 – 60 =
180 + 60 =
360 – 60 =
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.
21. Find 𝑠𝑖𝑛(120°)
23.
22. Find 𝑐𝑜𝑠(240°)
Find the below values. Use the circles to help you if necessary.
sin 30  _____
sin 45  _____
sin 60  _____
sin 90  _____
cos30  _____
cos 45  _____
cos 60  _____
cos90  _____
24.
What are the possible values for tangent? List them below, with the appropriate calculations.
25.
Find the following values WITHOUT looking at your unit circle on the previous page.
sin 60  _____
cos120  _____
tan180  _____
sin315  _____
cos90  _____
sin 240  _____
26.
Find the following values WITHOUT looking at your unit circle on the previous page.
tan 60  _____
cos 225  _____
sin  60   _____
tan390  _____
tan  315   _____
sin810  _____