Might Be Helpful...

For more help on the Unit Circle in general:

http://en.wikipedia.org/wiki/Unit_circle
http://www.mathsisfun.com/geometry/unit-circle.html

For more help on Radians:

http://en.wikipedia.org/wiki/Radians

Last Laughs...

Example #2

Sketch an angle of –45 in standard position. Then find the exact values of the cosine and the sine of –45 using a right triangle.

 Sketch the angle of measure –45 in standard position.

 

Step 2

 

Step 3

Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x–axis. The other leg is parallel to the y–axis.

The triangle contains angles of 45, 45, and 90.

 

Step 4

Find the hypotenuse of the triangle.

The hypotenuse is a radius of the unit circle.

hypotenuse = 1

Step 5

The triangle is a right isosceles triangle. Its two legs are equal.

Step 6

Find each leg of the triangle.

Each leg of the triangle is 1/(√ 2) times the hypotenuse.

Each leg

Step 7

Since the point lies in quadrant IV, its x–coordinate is positive and y–coordinate negative.

Step 8

Find the coordinates.

cos (–45) = (√ 2)/2

sin (–45) = –(√ 2)/2

The coordinates are

.

30-60-90 Triangle Review

Bob is at the fair and decides to ride the ferris wheel. He sits at the bottom of the ferris wheel and is 3 feet off the ground. The radius of the ferris wheel is 38 feet. The ferris wheel turns 300 degrees counter clockwise and stops to let someone new on. How far away from the ground would Bob be after the ferris wheel stops?
First, draw a picture:

The center of the wheel is A.
AB and AC are both 38 feet because they're both radii of the circle.
BD is 3 feet (height of Bob above the ground before 300 degree CCW rotation).
X=the height of Bob above the ground right now.
Angle CAE is 60 degrees (360-300=60)
X=ED

Bob's height off the ground is: X=41-AE
Since triangle AEC is a 30-60-90 triangle, AE must be half of AC.
and 38/2=19 feet.
AE=19 feet...fill it into the equation:
X=41-19, which reduces to
X=22
Bob is 22 feet off the ground after a 300 CCW rotation.

Answers

1.	180°, or π radians
2.	–90°, or  radians
3.	(0,1)
4.	(–1,0)
5.	(0,1)
6.	(1,0)
7.	(0,–1)
8.	(0,–1)
9.
10.
11.
12.
13.	(0,–1)
14.
15.	(–1,0)
16.
17.
18.
19.
20.
21.
22.	(0,1)
23.
24.
25.
26.
27.
28.
29.	(0,1)

Practice Questions

Practice Questions

1.	Give the angle between 0° and 360° that corresponds to the point (–1,0).
2.	What negative angle between 0° and – 360° corresponds to the point (0,–1)?

For problems 3 through 8, find the point on the unit circle that corresponds to the given angle.

3.
4.	540°
5.
6.	4π
7.	630°
8.	–90°

Find the point on the unit circle that corresponds to the following angles.

9.	–60°
10.
11.	420°
12.	–120°
13.
14.
15.	180°
16.

Find the point on the unit circle that corresponds to the given angle.

17.
18.	–30°
19.
20.	390°
21.
22.	–270°
23.	570°
24.

Find the point of the unit circle that corresponds to the given angle.

25.
26.	405°
27.	–45°
28.
29.	810°