Trigonometric Ratios Worksheet
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22Chapter 1 Trigonometric Ratios
1.1 Sine, Cosine, and Tangent of Special Angles
KEY CONCEPTS
• Exact trigonometric ratios for 30°, 45°, and 60° angles can be determined using special
triangles.
30°
2
3
45°
2
1
60°
45°
1
1
y
1
• Any point (x, y) on a unit circle can be joined to the
origin to form a radius 1 unit long.
• A rotation angle , in standard position, is formed by
(x, y)
proceeding counterclockwise from the initial arm on
1
the positive x-axis to the terminal arm through (x, y).
x
0
1
�1
• For any rotation angle, the reference angle is the acute
angle between the terminal arm and the x-axis.
• Given a point (x, y) on a unit circle, cos = x,
y
__
x .
sin = y, and tan =
�1
Example
Determine the exact values of the primary trigonometric ratios for 135°.
y
Solution
1
The measure of the reference angle is 180° −135°, or 45°.
P(x, y)
Use the special triangles to determine the sine and
cosine ratios for the reference angle.
135°
1
cos 45° = 1
___
___
sin 45° =
__
__
tan 45° = 1
2
√
√
2
x
0
1
�1
Since the terminal arm of a 135° angle in standard
position is in quadrant II, the x-coordinate of P
is negative.
1
cos 135° = − 1
___
___
�1
sin 135° =
__
tan 135° = −1
__
√
2
√
2
978-0-07-090893-2 Mathematics for College Technology 12 Study Guide and Exercise Book • MHR 1
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