This document provides examples and explanations of trigonometric functions including sine, cosine, and tangent. It discusses how to find the quadrant an image point lies in after a rotation about the origin. It also shows how to use trig functions to find the coordinates of a point on the unit circle after a rotation, and examples of evaluating trig functions with various angle measures. Finally, it gives an example of using trig functions to find the heights of the hour and minute hands of a clock at a certain time.

1. Section 4-3
Sines, Cosines, and Tangents

2. Warm-up
If (1, 0) is rotated the given amount about the origin, tell in
which quadrant its image lies.
a. 35° b. 210° c. 180°
I III On x-axis
d. 120° e. 325°
II IV

3. The Unit Circle: A circle whose center is on the origin and
radius is 1 unit
Ordered Pair on the Unit Circle: Under a rotation of
magnitude about the origin of the point (1, 0), the
image will consist of (cos , sin )
That is, R (1, 0) = (cos , sin )

4. Example 1
Evaluate the following.
3π
a. cos 270° b. cos
2
0 0
3π
c. sin 270° d. sin
2
1 1

5. Tangent
≠0
For all , as long as cos
sin
tan =
cos

6. Example 2
Evaluate the following.
a. tan 0 b. tan 270°
0 Undefined
Why?

7. Example 3
Find, to the nearest thousandth, the coordinates of the image
of (1, 0) under a rotation of
-10π
9
-10π -10π
(cos , sin )
9 9
≈ (-.940, .342)

8. Example 4
Give the sign of each of the following.
a. cos (-1°) b. sin (-1°) c. tan (-1°)
Positive Negative Negative

9. Example 5
A large clock with rotating hour and minute hands is on a
building with its center 20 feet tall. The length of the hour
hand is 1.5 feet and the length of the minute hand is 2 feet. At
8:20, give the height off the ground of the tip of each hand.
Minute hand: Hour hand:
-π
20 + 2sin( ) 20+ 1.5sin( -160°)
6
= 20 + 2(-.5) ≈ 19.49 feet
= 19 feet

10. Homework
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