1. Math 20-1 Chapter 2 Trigonometry
Teacher Notes
Math Chapter 2 Trigonometry
2.2 A Trig Ratios of Any Angle

2. 2.2A Trig Ratios of Any Angle (x, y, r)
Math Chapter 1 Sequences and Series
2.2A Trig Ratios of Any Angle (x, y, r)
Label the two special Triangles
300
450 2 1
600
450 1 1
2.2.1

3. Finding the Trig Ratios of an Angle in Standard Position
Suppose angle  is an angle in standard position.
Choose a point (x, y) on the terminal arm, at a distance r from the origin.
P(x, y) r y q x
r2 = x2 + y2
r = √x2 + y2
What is the relationship between x, y, and r?
2.2.2

4. Finding the Trig Ratios of an Angle in Standard Position
Suppose angle  is an angle in standard position. How are the ratios affected if we choose the point (-x, y) on the terminal arm?
P(-x, y) r y q
Ref -x
r2 = (-x)2 + y2
The horizontal and vertical lengths are considered as
directed distances.
2.2.3

5. Sine All ( -, + ) ( +, + ) Tangent Cosine ( -, - ) ( +, - )
How can you tell if the ratios will be positive or negative?
Sine
All
( -, + )
( +, + )
Tangent
Cosine
( -, - )
( +, - )
2.2.4

6. q Finding Trigonometric Ratios of Angles in Standard Position
Sine is positive.
All are positive.
Cos is positive.
Tan is positive.
2.2.5

7. Determine the sign of the ratio. 1. sin 1270 2. tan 240
The Cast Rule
Determine the sign of the ratio.
1. sin 1270     tan 240
3. cos tan 1450
5. cos  970     cos 450
7. sin cos 3150
Positive
Positive
Negative
Negative
Negative
Positive
Positive
Negative
2.2.6

8. The Cast Rule I II I IV I III II IV
i ) Sine ratios have positive values in quadrants _____ and _____ .
ii) Cosine ratios have positive values in quadrants _____ and _____ .
iii) Tangent ratios have positive values in quadrants _____ and _____ .
iv) sin > 0 and cos  < ______
v) tan  < 0 and sin  < ______ I IV I
III II IV
2.2.7

9. Finding the Exact Trig Ratios of an Angle in Standard Position
Given point P(5, 12) on the terminal arm calculate the exact values of the primary trig ratios.
r = √
P(5, 12)
r =13 13 12 q 5
2.2.8

10. Finding the Trig Ratios of an Angle in Standard Position
The point P(-2, 3) is on the terminal arm of q .in standard position.
Determine the exact value of the trigonometric ratios for angle q.
P(-2, 3) 3 q R -2
r2 = x2 + y2
r2 = (-2)2 + (3)2
r2 = 4 + 9
r2 = 13
r = √ 13
2.2.9

11. x -5 7 Challenging: Determine the exact value of the trig ratios given
Must be in Quad III x q -5 7
2.2.10

12. 1. sin 250 = 2. cos 1210 = 3. tan 3350 = 4. sin 00 = 5. tan 900 =
Calculator: Determine Approximate Trig Ratios (four decimal places)
1. sin 250 =
2. cos =
3. tan =
4. sin 00 =
5. tan 900 =
2.2.11

13. Assignment Suggested Questions Page 96:
1c, 2a,b, 3a,c, 4, 5d, 6 explain, 8a,b,e, 11a, 18b, 20
2.2.12