1. Trigonometric Ratios of Any Angle
Chapter 4 Trigonometric Functions
4.2
Trigonometric Ratios
of Any Angle
MATHPOWERTM 12, WESTERN EDITION
4.2.1

2. Trigonometric Ratios of Angles in Standard Position
When given an angle q in standard position and P(x, y), which
is any point on the terminal arm, the three primary ratios are
defined as follows:
By the Pythagorean
Theorem:
The three are defined as follows:
4.2.2

3. Finding Trigonometric Ratios of Angles in Standard Position
The point A(-4, 3) lies on the terminal arm of an angle q
in standard position. Determine the exact value of the
trigonometric ratios for angle q.
The reciprocal ratios follow from
the primary ratios:
4.2.3

4. Finding Trigonometric Ratios of Angles in Standard Position. . . .
4.2.4

5. Reference Angles 300 300 Principal angle _______
4.2.5

6. Reference Angles [cont’d]
Principal angle _______
Principal angle _______
Principal angle _______
Principal angle _______
4.2.6

7. Reference Angles Principal angle _______ Principal angle _______
4.2.7

8. Reference Angles Principal angle _______ Principal angle _______
4.2.8

9. Reference Angles and the CAST Rule
The trigonometric ratios of any angle can be written as the same
function of a positive acute angle called the reference angle. When
solving for angles greater than 900, the reference angle is used to
find the related trigonometric ratio. The is used to
determine the sign of the ratio.
Find the value of the following
trig ratios to 4 decimal places:
CAST Rule
a) sin 1500
b) sin 2100
Quadrant II
Quadrant I
Find the measure of q:
0 ≤ q < 3600
a) cos q =
The reference angle is
The angle q would be found in
Quadrant III
Quadrant IV
4.2.9

10. Exact Values of Special Angles 300, 600, and 900
sin 300 =
sin 600 =
300
cos 300 =
cos 600 =
600
tan 300 =
tan 600 =
450, 450, and 900
sin 450 =
450
cos 450 =
450
tan 450 =
4.2.10

11. Exact Values of Special Angles
4.2.11

12. Related Angles
4.2.12

13. Finding Exact Trigonometric Ratios State the ref angle and exact value
of each ratio:
CAST RULE
1. sin =
2. cos =
3. tan =
4. sin =
5. sin
6. cos
7. tan
8. tan
4.2.13

14. Finding Exact Trigonometric Ratios
1. sin =
2. cos =
3. cot =
4. csc =
5. csc
6. sec
7. cot
8. cot
4.2.14

15. Quadrantal Angles
(0, 1)
(-1, 0)
(1, 0)
(0, -1)
4.2.15

16. Finding Exact Trigonometric Ratios
1. sin 900 =
2. cos 900 =
3. sec =
4. csc =
5. csc
6. sec
7. cot
8. cot
4.2.16

17. Find Angle q ,Given an Exact Trigonometric Ratio
Find the value of angle q. State the answers in degrees and radians.
00 ≤ q < 3600
0 ≤ q < 2p
q =
q =
4.2.17

18. Find Angle q , Given an Exact Trigonometric Ratio
4.2.18

19. Assignment Suggested Questions: Pages 199-201 A 1-4, 7, 11 , 13-23, 43
B , 44-49, 53
4.2.19