Trigonometry
Trigonometry begins in the right
triangle, but it doesn’t have to be
restricted to triangles. The
trigonometric functions carry the
ideas of triangle trigonometry into a
broader world of real-valued
functions and wave forms.
1
Trigonometry Topics







Radian Measure
The Unit Circle
Trigonometric Functions
Larger Angles
Graphs of the Trig Functions
Trigonometric Identities
Solving Trig Equations
2
Radian Measure


To talk about trigonometric functions, it is
helpful to move to a different system of
angle measure, called radian measure.
A radian is the measure of a central angle
whose intercepted arc is equal in length to
the radius of the circle. r
s  r
s

3
Radian Measure



There are 2 radians in a full rotation -once around the circle
There are 360° in a full rotation
To convert from degrees to radians or
radians to degrees, use the proportion
degrees radians


360
2
4
Sample Problems

Find the degree
measure equivalent of
radians. 3
4
degrees radians


360
2
d
3 4


360
2
2d  270
d  135

Find the radian
measure equivalent
of 210°
degrees radians


360
2
210
r


360
2
360r  420
420 7
r

360
6
5
The Unit Circle


Imagine a circle on the
coordinate plane, with its
center at the origin, and
a radius of 1.
Choose a point on the
circle somewhere in
quadrant I.
6
The Unit Circle


Connect the origin to the
point, and from that point
drop a perpendicular to the
x-axis.
This creates a right triangle
with hypotenuse of 1.
7
The Unit Circle


The length of its legs
are the x- and ycoordinates of the
chosen point.
Applying the
definitions of the
trigonometric ratios to
this triangle gives
y
sin    y
1
 is the
angle of
rotation
x
cos    x
1
1
y
x
8
The Unit Circle

The coordinates of the chosen point are the
cosine and sine of the angle .


This provides a way to define functions sin()
and cos() for all real numbers .
y
sin(  )   y
1
The other trigonometric functions can be defined
from these.
x
cos( )   x
1
9
Trigonometric Functions
sin( )  y
1
csc( ) 
y
cos( )  x
1
sec( ) 
x
 is the
angle of
rotation
1
y
x
y
tan( ) 
x
x
cot( ) 
y
10
Around the Circle

As that point
moves around the
unit circle into
quadrants II, III,
and IV, the new
definitions of the
trigonometric
functions still
hold.
11
Reference Angles


The angles whose terminal sides fall in
quadrants II, III, and IV will have values of
sine, cosine and other trig functions which
are identical (except for sign) to the values
of angles in quadrant I.
The acute angle which produces the same
values is called the reference angle.
12
Reference Angles


The reference angle is the angle between the
terminal side and the nearest arm of the x-axis.
The reference angle is the angle, with vertex at
the origin, in the right triangle created by
dropping a perpendicular from the point on the
unit circle to the x-axis.
13
Quadrant II
Original angle


For an angle, , in
quadrant II, the
reference angle is   
In quadrant II,
sin() is positive
Reference angle  cos() is negative
 tan() is negative

14
Quadrant III
Original angle


Reference angle
For an angle, , in
quadrant III, the
reference angle is
-
In quadrant III,

sin() is negative
cos() is negative

tan() is positive

15
Quadrant IV

Reference angle

For an angle, , in
quadrant IV, the
reference angle is
2  
In quadrant IV,

sin() is negative
cos() is positive

tan() is negative

Original angle
16
All Seniors Take Calculus

Use the phrase “All Seniors Take Calculus”
to remember the signs of the trig functions
in different quadrants.
Seniors
All
Sine is positive All functions
are positive
Take
Tan is positive
Calculus
Cos is positive
17
Special Right Triangles
Angles measured in degrees:
1
sin 45  cos 45 
and tan 45  1
2
Angles measured in radians:
1
sin  / 4  cos  / 4 
and tan  / 4  1
2
18
Special Right Triangles
Angles measured in degrees:
1
sin 30  cos 60 
2
3
sin 60  cos30 
2
1
tan 60 
 3
tan 30
19
The 16-Point Unit Circle
20


1
0
1
1/2
3 /2
3 /3
2
2 3 /3
3
1
2 /2
1
2 /2
2
2
3 /2 1/2  3 2

3 /3
 3 /3  2
0
   1  
 1  0

1/2  3 
2 3 /3 2  3 /3
 3 /2 

 2 /2
1

 2 /2 
 1  2   2
 3 /2  3 /3  2 2

1/2 
 3 /3  3


 0  1  0    1
2 
1/2
2 3 /3
3

3
/2
3
/3





1
 2
1
  2 /2    2 
   2 /2
 3 /3 2
3 /3
1/2  3 2

 3 /2 

0


 1  0   1  


  3 /2 1/2  3 2 3 /3 2  3 /3

 2 
2

2
/2
2
/2
1
1







1/2  3 /2 

 3 /3 2 2 3 / 3  3
   1

1
0
0







0
Graphs of the Trig Functions

Sine


The most fundamental sine wave, y = sin(x),
has the graph shown.
It fluctuates from 0 to a high of 1, down to –1,
and back to 0, in a space of 2.
22
Graphs of the Trig Functions

The graph of y  a sin b( x  h)  k is
determined by four numbers, a, b, h, and k.




The amplitude, a, tells the height of each peak
and the depth of each trough.
The frequency, b, tells the number of full wave
patterns that are completed in a space of 2.
2
The period of the function is
b
The two remaining numbers, h and k, tell the
translation of the wave from the origin.
23
Sample Problem

5
4
3
2

1
2
1
1
1
2
Which of the following
equations best describes
the graph shown?
2


3
4
5


(A) y = 3sin(2x) - 1
(B) y = 2sin(4x)
(C) y = 2sin(2x) - 1
(D) y = 4sin(2x) - 1
(E) y = 3sin(4x)
24
Sample Problem
5

4
3
2

1
2
1
Find the baseline between
the high and low points.
1
1
2
2

3
5
y = 3sin(2x) - 1
Find height of each peak.

4

Graph is translated -1
vertically.
Amplitude is 3
Count number of waves in
2

Frequency is 2
25
Graphs of the Trig Functions

Cosine


The graph of y = cos(x) resembles the graph
of y = sin(x) but is shifted, or translated,
units to the left.
It fluctuates from 1
to 0, down to –1,
back to 0 and up to
1, in a space of 2.
26
Graphs of the Trig Functions

The values of a, b, h, and k change the shape
and location of the wave as for the sine.
y  a cos b( x  h)  k
Amplitude
Frequency
Period
Translation
a
b
2/b
h, k
Height of each peak
Number of full wave patterns
Space required to complete wave
Horizontal and vertical shift
27
Sample Problem

Which of the following
equations best describes
the graph?





(A) y = 3cos(5x) + 4
(B) y = 3cos(4x) + 5
(C) y = 4cos(3x) + 5
(D) y = 5cos(3x) + 4
(E) y = 5sin(4x) + 3
8
6
4
2
2
1
1
2
28
Sample Problem

Find the baseline


Vertical translation + 4
6
Find the height of
peak


8
Amplitude = 5
Number of waves in
2

4
2
2
1
1
2
y = 5 cos(3x) + 4
Frequency =3
29
Graphs of the Trig Functions

Tangent


The tangent function has a
discontinuous graph,
repeating in a period of .
Cotangent

Like the tangent, cotangent is
discontinuous.

Discontinuities of the
cotangent are  2 units left of
those for tangent.
30
Graphs of the Trig Functions

Secant and Cosecant


The secant and cosecant functions are the
reciprocals of the cosine and sine functions
respectively.
Imagine each graph is balancing on the peaks and
troughs of its reciprocal function.
31
Trigonometric Identities



An identity is an equation which is true for
all values of the variable.
There are many trig identities that are useful
in changing the appearance of an
expression.
The most important ones should be
committed to memory.
32
Trigonometric Identities

Reciprocal Identities
1
sin x 
csc x
1
cos x 
sec x

Quotient Identities
sin x
tan x 
cos x
cos x
cot x 
sin x
1
tan x 
cot x
33
Trigonometric Identities

Cofunction Identities

The function of an angle = the
cofunction of its complement.
sin x  cos(90  x)

sec x  csc(90  x)

tan x  cot(90  x)

34
Trigonometric Identities

Pythagorean Identities

The fundamental
Pythagorean identity
sin x  cos x  1

Divide the first by sin2x
1  cot x  csc x

Divide the first by cos2x
tan x  1  sec x
2
2
2
2
2
2
35
Trigonometric Identities
cos 2  cos 2   sin 2 
2sin  cos   sin(   )  sin(   )
cos 2  1  2sin 2 
cos 2  2cos 2   1
2cos cos   cos(   )  cos(   )
sin 2  2sin  cos 
2sin  sin   cos(   )  cos(   )
2 tan 
tan 2 
1  tan 2 
36
Trigonometric Identities
cos(   )  cos  cos   sin  sin 
cos(   )  cos  cos   sin  sin 
sin(   )  sin  cos   cos  sin 
sin( -  )  sin  cos   cos  sin 
tan   tan 
tan(   ) 
1  tan  tan 
tan   tan 
tan(   ) 
1  tan  tan 
37
Solving Trig Equations

Solve trigonometric equations by following
these steps:





If there is more than one trig function, use
identities to simplify
Let a variable represent the remaining function
Solve the equation for this new variable
Reinsert the trig function
Determine the argument which will produce the
desired value
38
Solving Trig Equations

To solving trig equations:

Use identities to simplify

Let variable = trig function

Solve for new variable

Reinsert the trig function

Determine the argument
39
Sample Problem

Solve
3  3 sin x  2 cos2 x  0
3  3 sin x  2 cos 2 x  0
3  3 sin x  2(1  sin 2 x)  0
1  3 sin x  2 sin 2 x  0
(1  2 sin x)(1  sin x)  0
1
sin x  or sin x  1
2
x  30 ,150 or x  90
40
Law of Sines and Cosines


All these relationships are based on the
assumption that the triangle is a right
triangle.
It is possible, however, to use trigonometry
to solve for unknown sides or angles in nonright triangles.
41
Law of Sines
a
b
c


sin( A) sin( B) sin(C)


In geometry, you learned that the largest angle of a
triangle was opposite the longest side, and the
smallest angle opposite the shortest side.
The Law of Sines says that the ratio of a side to the
sine of the opposite angle is constant throughout
the triangle.
42
Law of Sines

In ABC, mA = 38, mB = 42, and BC
= 12 cm. Find the length of side AC.



Draw a diagram to see the position of the given
angles and side.
BC is opposite A
You must find AC, the side opposite B.
C
A
B
43
Law of Sines

.... Find the length of side AC.

Use the Law of Sines with mA = 38, mB = 42,
and BC = 12
a
b

sin( A) sin( B)
12
b


sin(38 ) sin(42 )
12 sin 42  b sin 38
12 sin 42
b
sin 38
8.029
 13.041
13.042
44
Warning

The Law of Sines is useful when you know



the sizes of two sides and one angle or
two angles and one side.
However, the results can be ambiguous if the
given information is two sides and an angle
other than the included angle (ssa).
45
Law of Cosines


If you apply the Law of Cosines to a right triangle,
that extra term becomes zero, leaving just the
Pythagorean Theorem.
The Law of Cosines is most useful


when you know the lengths of all three sides and need to
find an angle, or
when you two sides and the included angle.
46
Law of Cosines

Triangle XYZ has sides of lengths 15, 22,
and 35. Find the measure of the angle C.
c 2  a 2  b 2  2ab cos(C)
15 C
22
35
35  15  22  2  15  22  cos(C)
2
2
2
1225  225  484  660 cos(C)
1225  709  660 cos(C)
47
Law of Cosines

... Find the measure of the largest angle of
the triangle.

516 660 cos(C )
15
22
35

516 
cos(C ) 
 .7818
660
1 

C  cos ( .7818)  1414
.
48
Laws of Sines and Cosines


A
Law of Sines:
a
b
c


sin A sin B sin C
b
C
c
B
a
Law of Cosines:
c  a  b  2ab cos C
2
2
2
49