Download Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle
4.4
Definition of Trig. Functions of Any
Angle
Let θ be an angle in standard position with 9x, y) a
point on the terminal side of θ and r = x 2  y 2 ≠ 0. (r is
always positive).
sin θ =
y
r
csc θ =
r
;y0
y
cos θ =
x
r
sec θ =
r
;x  0
x
tan θ =
y
;x  0
x
cot θ =
x
;y0
y
Example 1: Evaluating Trig. Functions
Let (-3, 4) be a point on the terminal side of θ. Find
the sine, cosine, and tangent of θ.
Quadrants
II
Sin θ = +
Cosθ = Tan θ = -
I
Sin θ = +
Cos θ = +
Tan θ = +
III
IV
Sin θ = Cos θ = Tan θ = +
Sin θ = Cos θ = +
Tan θ = -
Example 2: Evaluating Trig. Functions
5
Given    4 and cos θ > 0, find sin θ and sec θ.
Example 3: Trig. Functions of Quadrant
Angles
Evaluate the sine and cosine functions at the angles 0,
π/2, π, and 3π/2.
Reference Angles
Let θ be an angle in standard position. Its reference
angle is the acute angle θ´ formed by the terminal side of θ
and the horizontal axis.
θ
θ´
Example 4: Finding Reference Angles
Find the reference angle θ´.
A)Θ = 300°
B)Θ = 2.3
C)Θ = -135°
Evaluating Trig. Functions of Any Angle
To find the value of a trigonometric function of any
angle θ:
1. Determine the function value for the associated reference
angle θ´.
2. Depending on the quadrant in which θ lies, attach the
appropriate sign to the function value.
Example 5: Trig. Functions of
Nonacute Angles
Evaluate each trigonometric function.
A) Cos
B)
4
3
Tan (-210°)
C) Csc
11
4
Example 6: Using Trig. Identities
Let θ be an angle in quadrant II such that sin θ = 1/3.
Find cos θ by using trigonometric identities.