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Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
Objectives:
• Use the formula for the cosine of the difference of two angles.
• Use sum and difference formulas for cosines and sines.
• Use sum and difference formulas for tangents.
5.2 Sum and DifferenceFormulas
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The Cosine of the Difference of Two Angles
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Example 1: Using the Difference Formula for Cosines to Find an Exact Value
We know that
Obtain this exact value using
and the difference formula for cosines.
3cos30 .
2
cos30 cos 90 60
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Example1 Continued
•Using the difference formula for Cosines cos30 cos 90 60
cos30 cos 90 60
cos90 cos60 sin90 sin 60
1 30 1
2 2
32
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Example 2: Using the Difference Formula for Cosines to Find an Exact Value
•Find the exact value of cos70 cos40 sin 70 sin 40 . cos70 cos40 sin 70 sin 40
cos(70 40 )
cos30
32
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Example 3: Verifying an Identity
•Verify the identity: cos( )1 tan tan .
cos cos
cos cos sin sincos cos
cos cos sin sincos cos cos cos
cos coscos cos
sin sincos cos
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Example 3: (continued)•Continued
sin sin1
cos cos
1 tan tan
The identity is verified.
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Sum and Difference Formulas for Cosines and Sines
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Example 4: Using the Sine of a Sum to Find an Exact Value
Find the exact value of
using the fact that
5sin
12
5.
12 6 4
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Example 4 ContinuedUse the sum of sines formula
sin( ) sin cos cos sin
5sin sin
12 6 4
sin cos cos sin6 4 6 4
1 2 3 22 2 2 2
2 64
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Example 5 part A: Finding Exact Values
Suppose that for a quadrant II angle
and for a quadrant I angle
Find the exact value of
4sin
5
1sin
2 .
cos .
x
y
5r
4
x
2 2 2x y r
cosxr
2 2 24 5x 2 16 25x
2 9x 3x
3 35 5
4sin
5yr
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Example 5 Part B
Suppose that for a quadrant II angle
and for a quadrant I angle
Find the exact value of
4sin
5
1sin
2 .
cos .
x
y
2r
2 2 2x y r
cosxr
2 2 21 2x 2 1 4x
2 3x 3x
32
1sin
2yr
1
x
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Example 5 Part C
Suppose that for a quadrant II angle
and for a quadrant I angle
Find the exact value of
4sin
5
1sin
2 .
cos .
4sin
5
1sin
2 3
cos2
3cos
5
cos( ) cos cos sin sin 3 3 4 15 2 5 2
3 3 410 10
3 3 410
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Example 5: Part D
Suppose that for a quadrant II angle and
for a quadrant I angle Find the exact value
of
4sin
5
.
sin .
4sin
5
1sin
2 3
cos2
3cos
5
sin( ) sin cos cos sin 4 3 3 15 2 5 2
4 3 310 10
4 3 310
1sin
2
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Sum and Difference Formulas for Tangets
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Example 6: Verifying an Identity
Verify the identity: tan( ) tan . x x
tan tan1 tan tan
xx
tan 01 tan 0
xx
tan1x
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Example 7
Find the exact value of
= tan(20º + 100º)
= tan 120º
=
tan100tan201
tan100tan20
3
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Example 8If sin A = and A is in the third quadrant,
cos B = and B is in the fourth quadrant,
evaluate each of the following:
A) sin(A − B)
B) cos(A − B)
C) tan(A − B)
53
1312
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Example 8 Continuedsin(A − B) =
cos(A − B) =
tan(A − B) =
12
−3 5
−4
13
−5
sin A cos B − cos A sin B
3 512 45 13 5 13
65
2036
65
56
3 54 125 13 5 13
65
1548
65
33
tanBtanA1
tanBtanA
125
431
125
43
76
1116
33
56
cos A cos B + sin A sin B
