Download Section 4.4 Trigonometric Functions of Any Angle

Section 4.4
•
Trigonometric Functions of Any Angle
85
Name______________________________________________
Section 4.4 Trigonometric Functions of Any Angle
Objective: In this lesson you learned how to evaluate trigonometric
functions of any angle.
Define each term or concept.
Important Vocabulary
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.
I. Introduction (Pages 312−313)
Let θ be an angle in standard position with (x, y) a point on the
terminal side of θ and r = x 2 + y 2 ≠ 0 . Complete the
following definitions of the trigonometric functions of any angle:
What you should learn
How to evaluate
trigonometric functions
of any angle
y
sin θ =
y/r
cos θ =
x/r
tan θ =
y/x, x ≠ 0
cot θ =
x/y, y ≠ 0
sec θ =
r/x, x ≠ 0
csc θ =
r/y, y ≠ 0
Name the quadrants in which the sine function is positive.
Quadrants I and II
Name the quadrants in which the sine function is negative.
Quadrants III and IV
Name the quadrants in which the cosine function is positive.
Quadrants I and IV
Name the quadrants in which the cosine function is negative.
Quadrants II and III
Name the quadrants in which the tangent function is positive.
Quadrants I and III
Name the quadrants in which the tangent function is negative.
Quadrants II and IV
Example 1: If sin θ =
1
2
and tan θ < 0 , find cos θ .
− 0.8660
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
(x, y)
r
θ
x
86
Chapter 4
•
Trigonometry
II. Reference Angles (Page 314)
What you should learn
How to use reference
angles to evaluate
trigonometric functions
Example 2: Find the reference angle θ ′ for
(b) θ = 4.1
(a) θ = 210°
(a) 30°
(b) 0.9584 radians
III. Trigonometric Functions of Real Numbers
(Pages 315−317)
To find the value of a trigonometric function of any angle θ, . . .
What you should learn
How to evaluate
trigonometric functions
of real numbers
determine the function value for the associated reference angle
θ ′. Depending on the quadrant in which θ lies, affix the
appropriate sign to the function value.
Example 3: Evaluate sin
− 1/2
11π
.
6
Example 4: Evaluate cos 240° .
− 1/2
y
y
x
y
x
x
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.