Download 1 Additional Supplement Problems Trigonometry Reference Angles

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Additional Supplement Problems Trigonometry
Reference Angles
The reference angle is an angle between 0 o and 90 o (0 and

in radian measure) formed by the
2
terminal side of the given angle and the x-axis.
y
x
Example 1: Find the reference angles for the angles 250o and  389 o .
Solution:
2
█
Example 2: Find the reference angles for the angles
17
6
 radians and   radians.
11
7
Solution:
█
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Defining the Trigonometric Functions in Terms of Ordered Pairs
For the right triangle
recall that the the Pythagorean Theorem says that adj 2  opp 2  hyp 2 and that
cos  
adj
hyp
and
sin  
opp
hyp
We can use this idea to define the cosine and sine functions, and hence all six trigonometric
functions, for any point on the x-y coordinate plane as follows.
Suppose we are given the following diagram
y
x
We define the following six trigonometric function in terms of the ordered pair (x, y):
cos  
x
,
r
sin  
y
,
r
sec  
r
,
x
csc  
r
,
y
tan  
y
,
x
cot  
x
y
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We illustrate this idea in the next examples.
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Example 3: If sin  
12

, 0    , find cos , tan  , and sec  .
13
2
Solution
█
5 
Example 4: If sec    ,     , find sin  , cos , and cot  .
3 2
Solution
█
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Example 5: If tan  
7
, sin   0 , find sin  , cos , sec  , csc  , and cot  .
4
Solution:
█
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Recall that given cos and sin  ,
sec  
1
,
cos 
csc  
1
,
sin 
tan  
sin 
,
cos 
cot  
cos 
sin 
and
Pythagorean Identities
1. sin 2   cos 2   1
2. tan 2   1  sec 2 
3. cot 2   1  csc 2 
We can use these identities to simplify certain trigonometric expressions.
Example 6: Write each expression in terms of sines and/or cosines and simplify
a. cot( x ) sin( x )
b. sin( x )  cot( x ) cos( x )
Solution:
█
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Example 7: Write each expression in terms of sines and/or cosines and simplify
a.
sin( x )
 cos 2 ( x )
csc( x )
b.
1  cos( x ) tan( x ) csc( x )
csc( x )
c. (cos( x ) tan( x )  1)(sin( x )  1)
Solution:
█
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Amplitude, Period, and Phase Shift for the Sine and Cosine Functions
Recall the graph of y  sin( x ) and y  cos( x )
We can use this graph to graph related sine functions
The sine and cosine functions can be expressed in the general form as
y  a sin( bx  c )
and
y  a cos(bx  c)
To make functions of these forms easier to graph, we define the following terms.

Amplitude – gives the highest and lowest points of oscillation for the sine and cosine
function. The amplitude is given by the | a |

Period – smallest value on which the sine and cosine function repeats itself. The formula
2
for the period is given by
.
b

Phase shift – the amount on the x-axis the graph of the given sine and cosine function is
shifted horizontally from the graph of the standard graphs of y  sin( x ) and y  cos( x ) .
To find the phase shift, set bx  c  0 and solve for x. The value of x  
phase shift amount.
c
gives the
b
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Example 8: Determine amplitude, period, and phase shift for
y  3 sin( 2 x   )
Solution:
█
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Example 8: Determine amplitude, period, and phase shift for


y  2 cos(7  x  )
3

Solution:
█