Download Section 4.4 - Shelton State

Section 4.4
Trigonometric Functions of Any
Angle
Overview
• In this section we will find trigonometric
values for any angle.
• Doing so requires to consider rotations for
circles centered at the origin but not with a
radius of 1.
• We will use right triangle trigonometry to
write the appropriate ratios.
Definitions
y
sin  
r
x
cos  
r
y
tan  
x
r
csc  
y
r
sec  
x
x
cot  
y
Examples
1. The point (-9, 12) is on the terminal side
of an angle θ. Find the exact value of
each of the six trigonometric functions of
θ.
2. The point (-5, -4) is on the terminal side of
an angle θ. Find the exact value of each
of the six trigonometric functions of θ.
Trig Values And Their Signs
• Recall the signs of the coordinates of x and y
and each of the four quadrants:
• The six trigonometric values follow the same
rules and patterns.
Students All Take Calculus
Examples
1. Given that cos θ = -4/5 and θ is in
Quadrant II, find the remaining five
trigonometric values of θ.
2. Given that tan θ = 12/5 and cos θ < 0,
find the remaining five trigonometric
values of θ.
Reference Angles
•
•
1.
2.
3.
4.
A reference angle is a positive acute angle formed by
the terminal side and the x-axis.
To find a reference angle, first find find the coterminal
angle for the given angle that is between 0º and 360º
or 0 radians and 2π radians. Note the quadrant of the
coterminal angle.
If the coterminal angle is in Quadrant I, do nothing.
The coterminal angle is your reference angle.
If the coterminal angle is in Quadrant II, subtract it from
180º or π radians.
If the coterminal angle is in Quadrant III, subtract 180º
or π radians from the angle.
If the coterminal angle is in Quadrant IV, subtract it
from 360º or 2π radians.
Examples
• Find reference angles for each of the
following:
112
272
7
6

23
4