Math Analysis – Precalculus, Sullivan 10th Edition
Ch7Sec5Day1 1 3/1/2020
Section 7.5 – Sum and Difference Formulas – Day 1
This section continues the derivation of trigonometric identities by obtaining formulas that involve the sum or
difference of two angles, such as
cos( )
and
sin( ).
These formulas are called the sum and difference
formulas.
Use Sum and Difference Formulas to Find Exact Values
One use of the Sum and Difference formulas is to obtain the exact value of sine and cosine functions of an angle that
can be expressed as the sum or difference of angles whose sine and cosine are known exactly.
Theorem: Sum and Difference Formulas for Cosines
sinsincoscos)(cos
sinsincoscos)(cos
Example 1: Find the exact value of
.
12
7
cos
12
3
12
4
cos
12
7
cos
43
cos
4
sin
3
sin
4
cos
3
cos
by the theorem above
r
y
r
y
r
x
r
x
Now
2
3
,
2
1
)y,x(:
3
and
.
2
2
,
2
2
)y,x(:
4
1
2
2
1
2
3
1
2
2
1
2
1
2
2
2
3
2
2
2
1
4
6
4
2
62
4
1
Example 2: Find the exact value of
.20sin80sin20cos80cos
)2080(cos20sin80sin20cos80cos
by the Sum and Difference Formula Theorem
60cos
r
x
Now
.
2
3
,
2
1
)y,x(:60
1
2
1
2
1
Theorem: Sum and Difference Formulas for Sines
sincoscossin)(sin
sincoscossin)(sin
for Cosines
