5.4 Sum and Difference 5.4 Sum and Difference FormulasFormulas
Addition and Subtract of the Sine function
With the sine function the sign between the expressions stays the same.
vSinuCosvCosuSinvuSin
vSinuCosvCosuSinvuSin
Find the Sine of u + v
3045 vanduLet
4
26
4
2
4
63045
2
1
2
2
2
3
2
23045
304530453045
Sin
Sin
SinCosCosSinSin
Find the Sine of u - v
3045 vanduLet
4
26
4
2
4
63045
2
1
2
2
2
3
2
23045
304530453045
Sin
Sin
SinCosCosSinSin
Addition and Subtract of the Cosine function
With the cosine function the sign between the expressions change.
vSinuSinvCosuCosvuCos
vSinuSinvCosuCosvuCos
Find the Cosine of u + v
3045 vanduLet
4
26
4
2
4
63045
2
1
2
2
2
3
2
23045
304530453045
Cos
Cos
SinSinCosCosCos
Find the Cosine of u - v
3045 vanduLet
4
26
4
2
4
63045
2
1
2
2
2
3
2
23045
304530453045
Cos
Cos
SinSinCosCosCos
Addition and Subtract of the Tangent function
Both are Rational expressions
vu
vuvu
vu
vuvu
tantan1
tantantan
tantan1
tantantan
Find the Tangent of u + v
3045 vanduLet
33
11
33
13045tan
30tan45tan1
30tan45tan3045tan
Find the Tangent of u + v
3045 vanduLet
33
11
33
13045tan
30tan45tan1
30tan45tan3045tan
326
3612
33
33
33
33
33
33)3045tan(
Find the Tangent of u - v
3045 vanduLet
33
)1(1
33
13045tan
30tan45tan1
30tan45tan3045tan
tantan1
tantantan
vu
vuvu
Find the Tangent of u - v
3045 vanduLet
33
)1(1
33
13045tan
30tan45tan1
30tan45tan3045tan
tantan1
tantantan
vu
vuvu
326
3612
33
33
33
333045tan
33
333045tan
Simplify
Remember: sin(u-v)=sin u·cos v-cos u·sin v
36
7
36
7
36
7 SinCosCosSinSin
Simplify
Remember: sin(u-v)=sin u·cos v-cos u·sin v
2
3
2
3
2
1
2
1
36
7
36
7
36
7
36
7
Sin
SinCosCosSinSin
Simplify
Remember: sin(u-v)=sin u·cos v-cos u·sin v
2
1
4
2
4
3
4
1
2
3
2
3
2
1
2
1
36
7
36
7
36
7
36
7
Sin
SinCosCosSinSin
Break up the angle and find Sin, Cos and Tan
Leave as a Rational Expression
12
11
Break up the angle and find Sin, Cos and Tan
Leave as a Rational Expression
64
3
12
11
Break up the angle and find Sin, Cos and Tan
Leave as a Rational Expression
64
3
64
3
64
3sin
SinCosCosSin
Break up the angle and find Sin, Cos and Tan
Leave as a Rational Expression
2
1
2
2
2
3
2
2
64
3
64
3
64
3sin
SinCosCosSin
Break up the angle and find Sin, Cos and Tan
Leave as a Rational Expression
4
26
4
2
4
6
2
1
2
2
2
3
2
2
64
3
64
3
64
3sin
SinCosCosSin
Break up the angle and find Sin, Cos and Tan
Leave as a Rational Expression
4
26
4
2
4
6
2
1
2
2
2
3
2
2
64
3
64
3
64
3cos
SinSinCosCos
Break up the angle and find Sin, Cos and Tan
Tan can be done two ways.
32
31
33
33
11
33
1
6tan
43
tan1
6tan
43
tan
64
3tan
3226
26
426
426
643
643
64
3
Cos
SinTan
Sin u = 5/13
What is Cos u ?
Sin u = 5/13
What is Cos u ?
13
12;
169
144cos
169
144
13
51cos
1cos13
5
1cossin
2
22
22
22
uu
u
u
uu
Find cos(arc cos x – arc tan x)
Let u = arc cos x
v = arc tan x
u
x
121 x
v
x12 x
1
Find cos(arc cos x – arc tan x)
u
x
121 x
v
x12 x1
1cos
1sin
cos
1sin
2
2
2
xv
x
xv
xu
xu
1
Find cos(arc cos x – arc tan x)
1
1cos
1sin
cos
1sin
2
2
2
xv
x
xv
xu
xu
1
1
1
1
1
11
1
1
sinsincoscoscos
2
2
2
2
2
2
2
2
x
xx
x
x
x
x
x
xx
xx
vuvuvu
HomeworkHomework
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31, 42, 53, 60,31, 42, 53, 60,
71, 7471, 74
Homework
Page 384 – 385
# 4,10, 20, 28,
37, 47, 59,
62, 70, 73
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