Math 20-1 Chapter 2 Trigonometry
2.2 A Trig Ratios of Any Angle
Teacher Notes
Math 20-1 Chapter 1 Sequences and Series
2.2A Trig Ratios of Any Angle (x, y, r)
Label the two special Triangles
300
2
3
1
2
450
450
600
1
1
2.2.1
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.
r
P(x, y)
y
sin  
r
y
x
cos  
r

x
r2
x2
y2
= +
r = √x2 + y2
y
tan  
x
What is the relationship between x, y, and r?
2.2.2
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?
y
sin  
r
P(-x, y)
r
y
Ref
-x

x
cos   
r
y
tan   
x
r2 = (-x)2 + y2
The horizontal and vertical lengths are considered as
directed distances.
2.2.3
How can you tell if the ratios will be positive or negative?
( -, + )
y
r
cos  
All
Tangent
Cosine
( +, + )
( -, - )
sin  
Sine
( +, - )
x
r
tan  
y
x
2.2.4
Finding Trigonometric Ratios of Angles in Standard Position
y
sin  
r
cos   
y
sin  
r
x
r
y
tan   
x
Sine is positive.
cos  

x
r
y
tan  
x
All are positive.
y
sin   
r
sin   
x
cos   
r
x
cos  
r
y
tan  
x
tan   
Tan is positive.
y
r
y
x
Cos is positive.
2.2.5
The Cast Rule
Determine the sign of the ratio.
1.
sin 1270 Positive
2.
tan 240
Positive
3.
cos 2600
Negative
4.
tan 1450
Negative
5.
cos 970
Negative
6.
cos 450
Positive
7.
sin 3140
Negative
8.
cos 3150
Positive
2.2.6
The Cast Rule
I and _____
II .
i ) Sine ratios have positive values in quadrants _____
I and _____
IV .
ii) Cosine ratios have positive values in quadrants _____
III .
I and _____
iii) Tangent ratios have positive values in quadrants _____
iv) sin > 0 and cos  < 0
II
______
v) tan  < 0 and sin  < 0
IV
______
2.2.7
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 = √52 + 122
r =13
P(5, 12)
13

5
12
12
y
sin  
sin  
13
r
x
5
cos  
r cos  13
y
12
tan  
x tan   5
2.2.8
Finding the Trig Ratios of an Angle in Standard Position
The point P(-2, 3) is on the terminal arm of  .in standard position.
Determine the exact value of the trigonometric ratios for angle .
3
sin  
13
P(-2, 3)
13
3
R
-2
r2
x2
y2
= +
r2 = (-2)2 + (3)2
r2 = 4 + 9
r2 = 13
r = √ 13

2
cos   
13
3
tan   
2
2.2.9
Challenging: Determine the exact value of the trig ratios given
5
sin    , tan  0
7

Must be in Quad III
sin   
cos  
tan  
x

-5
7
5
7
2 6
7
5
2 6
x 2  72  52
x   72  52
x   24
x  2 6
2.2.10
Calculator: Determine Approximate Trig Ratios (four decimal places)
1. sin 250 = 0.4226
2. cos 1210 = 0.5150
3. tan 3350 = 0.4663
4. sin 00 = 0
5. tan 900 = undefined
2.2.11
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1c, 2a,b, 3a,c, 4, 5d, 6 explain, 8a,b,e, 11a, 18b, 20
2.2.12