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Supplemental Section: Trigonometry
Angle Measurement
Angles can be measured in degrees.
Counterclockwise Angles
Clockwise Angles
y
y
x
Important Degree Relationships
1 revolution = 360 0
3
revolution = 270 0
4
1
revolution = 1800
2
1
revolution = 90 0
4
Angles can also be measured in radians.
x
y
x
2
Consider the unit circle
y
x
Important Radian Relationships
1 revolution = 2 radians
3
3
revolution =
radians
4
2
1
revolution =  radians
2
1

revolution =
radians
4
2
y
x
3
Example 1: Draw, in standard position, the angle 330 0
Solution:
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Example 2: Draw, in standard position, the angle
5
.
4
Solution:
█
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Angle Measurement Conversions
Formula for converting from degree to radian measure
  
Radian measure  (Degree measure)  

 180 
Formula for converting from degree to radian measure
 180 
Degree measure  (Radian measure)  

  
Example 3: Convert 1200 from degrees to radians.
Solution:
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Example 4: Convert 
7
from radians to degrees.
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Solution: Using the formula
 180 
Degree measure  (Radian measure)  

  
we obtain the result
Degree measure  
7  180 
1260

 420 0

3   
3
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The Sine and Cosine Functions
We can define the cosine and sine of an angle  in radians, denoted as cos and sin  ,
as the x and y coordinates respectively of a point P on the unit circle that is determined by
the angle  .
Consider the unit circle
y
x
Sine and Cosine of Basic Angle Values
 Degrees
0
30
45
60
90
180
270
360
 Radians
0

6

4

3

2

3
2
2
cos
sin 
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Note: If we know the cosine and sine values for an angle  in the first quadrant, we can
determine the sine and cosine of related angles in other quadrants.
Sign Diagram for Cosine and Sine Values
y
x
7
Example 5: Compute the exact values of the sine and cosine for the angle
4
3
Solution:
█
Example 6: Compute the exact values of the sine and cosine for the angle
11
.
4
Solution:
█
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Other Trigonometric Functions
Tangent: tan  
Secant: sec 
Cosecant: csc 
Cotangent: cot  
Example 7: Find the exact values of the six trigonometric functions for the angle
11
.
6
Solution:
█
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Note: sin 2   (sin  ) 2 , cos 2   (cos  ) 2
Basic Trigonometric Identities
Pythagorean Identities
1. sin 2   cos 2   1
2. tan 2   1  sec 2 
3. cot 2   1  csc 2 
Double Angle Formula
sin 2  2 sin  cos
Example 8: Find the values of x in the interval [0,2 ] that satisfy the equation
sin 2x  sin x .
Solution: We solve the equation using the following steps:
sin 2 x  sin x
2 sin x cos x  sin x
(Re place left hand side with sin 2 x  2 sin x cos x)
2 sin x cos x  sin x  0
(Subtract sin x from both sides)
sin x(2 cos x  1)  0
(Factor sin x)
sin x  0, 2 cos x  1  0
(Set each factor equal to zero)
sin x  0, 2 cos x  1
(Solve for cos x)
sin x  0, cos x 
1
2
In the interval [0,2 ] , sin x  0 when x  0,  , 2 . In the interval [0,2 ] , cos x 
x
 5
3
,
3
1
when
2
. Thus the five solutions are
x  0,

3
, ,
5
, 2
3
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Graphs of Sine and Cosine
Period – distance on the x axis required for a trigonometric function to repeat its output
values.
The period of f ( x)  sin x and f ( x)  cos x is 2 .
Example 9: Graph f ( x)  sin x
Solution: The graph can be plotted using the following Maple command:
> plot(sin(x), x = -4*Pi..4*Pi, y = -2..2, color = red, thickness = 2);
█
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Example 10: Graph f ( x)  cos x
Solution: The graph can be plotted using the following Maple command:
> plot(cos(x), x = -4*Pi..4*Pi, y = -2..2, color = red, thickness = 2);
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Graph of the Tangent Function
Period of the tangent function is  radians. The vertical asymptotes of the tangent function
sin x

f ( x) 
are the values of x where cos x  0 , that is the odd multiples of
,
cos x
2
3   3

, , , , .
2
2 2 2
Example 11: Graph f ( x)  tan x .
Solution: The graph is given by the following:
█
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Practice Problems
1. Convert from degrees to radians.
a. 210 0
c. 9 0
b.  315 0
d. 36 0
2. Convert from radians to degrees.
a. 4
b. 
c. 
7
2
d.
3
8
8
3
3. Draw, in standard position, the angle whose measure is given.
3
a.  315 0
c. 
rad
4
7
b.
rad
d.  3 rad
3
4. Find the exact trigonometric ratios for the angle whose radian measure is given.
3
4
a.
c.
4
3
11
b. 
d.
6
5. Find all values of x in the interval [0, 2 ] that satisfy the equation.
a. 2 cos x  1  0
b. sin 2x  cos x
Selected Answers
1. a.
7
7
, b. 
6
4
2. a. 720 0 , b.  630 0
3
2
3
2
3
3
3
3
 1 , sec
  2 , csc
 2 , cot
 1


, sin
, tan
4
2
4
4
4
4
4
2
b. cos   1 , sin   0 , tan   0 , sec   1 , csc  : undefined , cot  : undefined
4. a. cos
5. x 
 5
3
,
3