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Special Angles and the CAST RULE
Many real­world applications of trigonometry deal with angle measurements of 30, 45,
and 60 degrees in a right triangle. ie) Construction.
Because we see these angles so much it is worth memorizing their EXACT trigonometric
values!
Special Triangles:
To help learn these values, try memorizing the following special triangles!
60
45
2
1
1
45
30
1
Example: Using the special triangles provide EXACT answers for the following.
θ
0
30
45
60
90
sinθ
cosθ
tanθ
Related Angles:
y
Terminal Arm
Rotational Angle θ
Related/Reference Angle
x
Rotational Angle θ ­ The angle formed between the positive x­axis,
counterclockwise to the terminal arm.
Related/Reference Angle ­ the acute angle between the teminal arm and the x­axis.
Example: Sketch the following angles in standard position, and state
the corresponding RELATED angle.
o
a) 240
c) 330
o
b) 135
o
o
d) 210
CAST rule:
The CAST rule is used to determine the "sign" of a trigonometric ratio,
depending on the location of the terminal arm. Each letter determines
the POSITIVE ratio.
Quadrant II
y
S
(sine)
Quadrant I
A
(all)
x
T
(tan)
Quadrant III
C
(cos)
Quadrant IV
Evaluating Trig Ratios using RELATED angles and the CAST rule:
Since Trigonometric Functions are PERIODIC, the same trig ratios, repeat at
regular intervals!!!
Because of this we can use RELATED angles to evaluate trigonometric ratios
larger than 90 degrees.
Steps:
Example:
Find the exact value of sin210o
1.) Rewrite the trig ratio
using the RELATED angle.
1.) sin210o = sin30o
2.) sin 30o =
2.) Using the special triangles,
state the EXACT value.
1
2
3.) Since 210o is in Quadrant III,
3.) Use the CAST rule to
determine the sign.
(sine is Negative)
o
∴ sin210 = ­ 1
2
Example: Find the EXACT Value of each trigonometric function.
a) cos300o
b) tan120o
c) sin330o
d) tan225o
Homework: Worksheet
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