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Section 6.3
Trigonometric
Functions of Any
Angle
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives
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Find angles that are coterminal with a given angle and
find the complement and the supplement of a given
angle.
Determine the six trigonometric function values for any
angle in standard position when the coordinates of a
point on the terminal side are given.
Find the function values for any angle whose terminal
side lies on an axis.
Find the function values for an angle whose terminal
side makes an angle of 30º, 45º, or 60º with the x-axis.
Use a calculator to find function values and angles.
Angle
An angle is the union of two rays with a common
endpoint called the vertex. We can think of it as a
rotation. Locate a ray along the positive x-axis with its
endpoint at the origin. This ray is called the initial side of
the angle. Now rotate a copy of this ray. A rotation
counterclockwise is a positive rotation and rotation
clockwise is a negative rotation. The ray at the end of
the rotation is called the terminal side of the angle. The
angle formed is said to be in standard position.
Angle
Angle
The measure of an angle or rotation may be given in
degrees. One complete positive revolution or rotation has
a measure of 360º. One half of a revolution has a
measure of 180º …
Angle
One fourth of a revolution has a measure of 90º, and so
on.
Angle
Angle measure of 60º, 135º, 330º, and 420º have
terminal sides that lie in quadrants I, II, IV and I
respectively.
Angle
The negative rotations –30º, –110º, and –225º represent
angles with terminal sides in quadrants IV, III, and II
respectively.
Coterminal Angles
If two or more angles have the same terminal side, the
angles are said to be coterminal. To find angles
coterminal with given angles, we add or subtract multiples
of 360º.
Example
Find two positive angles and two negative angles that are
coterminal with (a) 51º (b) –7º.
Solution:
a) Add or subtract multiples of 360º. Many answers are
possible.
51º + 360º = 411º
51º + 3(360º) = 1131º
Example (cont)
51º – 360º = –309º
b) We have the following:
–7º + 360º = 353º
–7º – 360º = –367º
51º – 2(360º) = –669º
–7º + 2(360º) = 713º
–7º – 10(360º) = –3607º
Classification of Angles
Complementary Angles
Two acute angles are complementary if their sum is 90º.
For example, angles that measure 10º and 80º are
complementary because 10º + 80º = 90º.
Supplementary Angles
Two positive angles are supplementary if their sum is
180º. For example, angles that measure 45º and 135º are
supplementary because 45º + 135º = 180º.
Example
Find the complement and supplement of 71.46º.
Solution:
90º 71.46º  18.54º
180  71.46º  108.54º
The complement of 71.46º is 18.54º and the
supplement of 71.46º is 108.54º.
Trigonometric Functions of Angles
Consider a right triangle with one
vertex at the origin of a coordinate
system and one vertex on the
positive x-axis. The other vertex P, a
point on the circle whose center is at
the origin and whose radius r is the
length of the hypotenuse of the
triangle. This triangle is a reference triangle for angle
, which is in standard position. Note that y is the
length of the side opposite  and x is the length of the
side adjacent to .
Trigonometric Functions of Angles
The three trigonometric functions of  are defined as
follows:
opp y
opp y
adj
x
sin 

tan 

cos 

hyp r
adj x
hyp r
Since x and y are the coordinates of the point P and
the length of the radius is the hypotenuse, we have:
y-coordinate
sin  
y-coordinate
radius
tan  
x-coordinate
x-coordinate
cos 
radius
Trigonometric Functions of Angles
We will use these definitions for functions of angles of
any measure.
Trigonometric Functions of Any Angle 
Suppose that P(x, y) is any point other than the vertex on
the terminal side of any angle  in standard position, and
r is the radius, or distance from the origin to P(x,y). Then
the trigonometric functions are defined as follows:
radius
r
y-coordinate y
csc 

sin  

y-coordinate y
radius
r
x-coordinate x
cos 

radius
r
y-coordinate y
tan  

x-coordinate x
radius
sec 

x-coordinate
x-coordinate
cot  

y-coordinate
r
x
x
y
Example
Find the six trigonometric
function values for the angle
shown.
Solution:
Determine r, distance from (0, 0) to (–4, –3).
r
x  0   y  0  
r
4   3
2
2
2
2
x 2  y2
 16  9  25  5
Example (cont)
Substitute –4 for x, –3 for y,
and 5 for r.
y 3
3
sin   

r
5
5
r
5
5
csc  

y 3
3
x 4
4
cos  

r
5
5
r
5
5
sec  

x 4
4
y 3 3
tan   

x 4 4
x 4 4
cot   

y 3 3
Example
2
Given that tan   
and  is in the second quadrant,
3
find the other function values.
Solution:
y
2
2
Sketch a second-quadrant angle using tan     
x
3 3
hyp  2 2  32  13
Example (cont)
Use the lengths of the three sides to find the appropriate
ratios.
2
2 13
sin  

13
13
13
csc 
2
3
3 13
cos  

13
13
13
sec  
3
2
tan   
3
3
cot   
2
Terminal Side on an Axis
An angle whose terminal side falls on one of the axes is a
quadrantal angle. One of the coordinates of any point on
that side is 0. The definitions of the trigonometric
functions still apply, but in some cases, function values
will not be defined because a denominator will be 0.
Example
Find the sine, cosine, and tangent values for 90º, 180º,
270º, and 360º.
Solution:
Sketch the angle in standard position, label a point on
the terminal side, choosing (0, 1).
1
sin 90º   1
1
0
cos 90º   0
1
1
tan 90º  Not defined
0
Example (cont)
0
sin180º   0
1
1
cos180º 
 1
1
0
tan180º 
0
1
1
sin 270º 
 1
1
0
cos 270º   0
1
1
tan 270º 
Not defined
0
Example (cont)
0
sin 360º   0
1
1
cos 360º   1
1
0
tan 360º   0
1
Reference Angles: 30º, 45º, 60º)
Consider the angle 150º, its terminal side makes a 30º
angle with the x-axis.
1
sin150º  sin 30º 
2
3
cos150º   cos 30º  
2
tan150º   tan 30º
1
3


3
3
Example
Find the sine, cosine, and tangent values for each of the
following:
a) 225º
b) –780º
Solution:
Draw the figure,
terminal side 225º,
reference angle is
225º – 180º = 45º
Example (cont)
2
sin 225º  
2
2
cos 225º  
2
tan 225º  1
Example (cont)
Draw the figure, terminal side
–780º is coterminal with
–780º + 2(360º) = –60º,
reference angle is 60º.
Example (cont)
3
sin 780º   
2
1
cos 780º  
2
tan 780º    3
Example
Given the function value and the quadrant restriction, find .
a) sin  = 0.2812, 90º <  < 180º
b) cot  = –0.1611, 270º <  < 360º
Solution:
Sketch the angle in the second quadrant.
Use a calculator to find the
acute (reference) angle
whose sine is 0.2812. It’s
approximately 16.33º. Now
180º – 16.33º = 163.37º.
Example (cont)
b) cot  = –0.1611, 270º <  < 360º
Sketch the angle in the fourth quadrant.
1
1
tan  

 6.2073
cot  0.1611
Use a calculator to find the
acute (reference) angle
whose tangent is –6.2073.
It’s approximately 80.85º.
Now 360º – 80.85 = 279.15º.
Bearing: Second-Type
In aerial navigation, directions, or bearings, are given in
degrees clockwise from north. Thus east is 90º, south is
180º, and west is 270º.
Example
An airplane flies 218 mi
from an airport in a
direction of 245º. How
far south of the airport
is the plane then? How
far west?
Solution:
Sketch a diagram.
Example (cont)
Find the measure of angle
ABC:
B  270º 245º  25º
Find how far south the plane is, that is, the length b:
b
 sin 25º
218
b  218sin 25º  92 mi
Example (cont)
Find how far west the plane
is, that is, the length a:
a
 cos 25º
218
a  218 cos 25º  198 mi
The airplane is about 92 mi south and about 198 mi
west of the airport.