40
Sum and Difference Identities
In this section, you will learn how to apply identities involving the sum or
difference of two variables.
Formulas for sin (x + y) and sin (x − y)
Let x and y be two angles as shown in Figure 115.
Figure 115
Let A be the point on the x-axis such that |OA| = 1. From A drop the
perpendicular to the terminal side of x. From B drop the perpendicular to
the x-axis. Then
Area ∆OAB = Area ∆OAC + area∆OCB.
But
1
1
Area ∆OAC = |OC||AC| = sin x cos x.
2
2
1
1
Area ∆OCB = |OC||BC| = |OB|2 sin y cos y.
2
2
1
1
Area ∆OAB = |BD||OA| = |OB| sin (x + y).
2
2
Hence,
1
1
1
|OB| sin (x + y) = sin x cos x + |OB|2 sin y cos y.
2
2
2
197
2
and using the fact that |OB| =
Mutliplying both sides by |OB|
obtains the addition formula for the sine function:
cos x
cos y
one
sin (x + y) = sin x cos y + cos x sin y.
To find the difference formula for the sine function we proceed as follows:
sin (x − y) =
sin (x + (−y))
= sin x cos (−y) + cos x sin (−y)
=
sin x cos y − cos x sin y
where we use the fact that the sine function is odd and the cosine function
is even.
Example 40.1
Find the exact value of sin 75◦ .
Solution.
Notice first that 75◦ = 30◦ + 45◦ . Thus,
sin 75◦ =
sin (45◦ + 30◦ )
◦
= sin 45
cos 30√◦ + cos√45◦√sin 30◦
√ √
2 3
=
+ 22 12 = 6+4 2
2 2
Example 40.2
π
.
Find the exact value of sin 12
Solution.
π
Since 12
=
π
4
− π3 , the difference formula for sine gives
π
sin 12
=
sin ( π4 − π6 )
π
= √
sin√π4 cos√π6 − cos√π4 sin
√ 6
= 22 23 − 22 12 = 6−4 2
Example 40.3
Show that cos ( π2 − x) = sin x using the difference formula of the sine function.
198
Solution.
Since the sine function is an odd function then we can write
sin x =
− sin (−x) = − sin [( π2 − x) − π2 ]
= −[sin ( π2 − x) cos ( π2 ) − cos ( π2 − x) sin ( π2 )]
=
cos ( π2 − x)
Theorem 40.1 (Cofunctions Identities)
For any angle x, measured in radians, we have
sin ( π2 − x) = cos x
sec ( π2 − x) = csc x
tan ( π2 − x) = cot x
cos ( π2 − x) = sin x
csc ( π2 − x) = sec x
cot ( π2 − x) = tan x
Proof.
Recall that sin ( π2 ) = 1 and cos ( π2 ) = 0.
sin ( π2 − x)
cos ( π2 − x)
sec ( π2 − x)
csc ( π2 − x)
= sin ( π2 ) cos x − cos ( π2 ) sin x = cos x
=
sin x (See Example 16.3)
1
=
= sin1 x = csc x
cos ( π2 −x)
1
= cos1 x = sec x
=
sin ( π −x)
tan ( π2 − x) =
cot ( π2 − x) =
2
sin ( π2 −x)
x
= cos
= cot x
cos ( π2 −x)
sin x
1
1
= cot x = tan x
tan ( π2 −x)
Formulas for cos (x + y) and cos (x − y)
Since sin x and cos x are cofunctions of each other then
cos (x + y) = sin ( π2 − (x + y)) = sin ( π2 − x) − y
= sin ( π2 − x) cos y − cos ( π2 − x) sin y
=
cos x cos y − sin x sin y
For the difference formula we have
cos (x − y) =
cos (x + (−y))
= cos x cos (−y) − sin x sin (−y)
=
cos x cos y + sin x sin y
where we have used the fact that the sine function is odd and the cosine is
even.
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Example 40.4
.
Find the exact value of cos 7π
12
Solution.
cos 7π
=
cos ( π4 + π3 )
12
π
π
π
= √
cos 4 cos
− sin√π4 sin
√ 3√
√ 3
= 22 12 − 22 23 = 2−4 6
Example 40.5
Find the exact value of: sin 42◦ cos 12◦ − cos 42◦ sin 12◦ .
Solution.
sin 42◦ cos 12◦ − cos 42◦ sin 12◦ = sin (42◦ − 12◦ ) = sin 30◦ = 21 .
Example 40.6
√
Suppose that α and β are both in the third quadrant and that sin α = − 23
and sin β = − 12 . Determine the value of cos (α + β).
Solution.
p
Since α and β are in the third √quadrant then cos α = − 1 − sin2 α = − 12
p
and cos β = − 1 − sin2 β = − 23 . Thus,
cos (α + β) =
cos α
cos β − √sin α sin β
√
3
1
= (− 2 )(− 2 ) − (− 23 )(− 12 ) = 0
Formulas for tan (x + y) and tan (x − y)
Using the sum formulas for the sine and the cosine functions we have
tan (x + y) =
=
=
sin (x+y)
cos (x+y)
sin x cos y+cos x sin y
cos x cos y−sin x sin y
sin x cos y
cos x sin y
+ cos
cos x cos y
x cos y
=
sin x sin y
1− cos
x cos y
tan x+tan y
1−tan x tan y
For the difference formula we have
tan x+tan (−y)
tan (x − y) = tan (x + (−y)) = 1−tan
x tan (−y)
tan x−tan y
=
1+tan x tan y
since tan (−x) = − tan x.
200
Example 40.7
Establish the identity: tan (θ + π) = tan θ.
Solution.
tan (θ + π) =
tan θ+tan π
1−tan θ tan π
= tan θ since tan π = 0.
201
Review Problems
Exercise 40.1
Find the exact value of the expression
◦
(a) sin (45◦ + 30
).
π
π
(b) cos 4 − 3 .
(c) tan π6 + π4 .
Exercise 40.2
Find the exact value of the expression
(a) cos 212◦ cos 122◦ + sin 212◦ sin 122◦ .
(b) sin 167◦ cos 107◦ − cos 167◦ sin 107◦ .
Exercise 40.3
Find the exact value of the expression
(a) sin 5π
cos π4 − cos 5π
sin π4 .
12
12
π
π
π
(b) cos 12 cos 4 − sin 12 sin π4 .
Exercise 40.4
Find the exact value of the expression
(a)
(b)
tan 7π
−tan π4
12
.
1+tan 7π
tan π4
12
π
π
tan 6 +tan 3
.
1−tan π6 tan π3
Exercise 40.5
Write each expression in terms of a single trigonometric function.
(a) sin x cos 3x + cos x sin 3x.
(b) sin 7x cos 3x − sin x sin 5x.
Exercise 40.6
Write each expression in terms of a single trigonometric function.
(a) cos 4x cos (−2x) − sin 4x sin (−2x).
tan 3x+tan 4x
(b) 1−tan
.
3x tan 4x
tan 2x−tan 3x
(c) 1+tan 2x tan 3x .
202
Exercise 40.7
8
, α in Quadrant I, and sin β = − 17
, β in Quadrant II, find
Given tan α = 24
7
the exact value of
(a) sin (α + β)
(c) tan (α − β).
(b) cos (α + β)
Exercise 40.8
12
Given sin α = − 45 , α in Quadrant III, and cos β = − 13
, β in Quadrant II,
find the exact value of
(a) sin (α − β)
(b) cos (α + β)
(c) tan (α + β).
Exercise 40.9
Given cos α = − 35 , α in Quadrant III, and sin β =
the exact value of
(a) sin (α − β)
(b) cos (α + β)
5
,
13
β in Quadrant I, find
(c) tan (α + β).
Exercise 40.10
Establish the following identities:
(a) sin θ + π2 = cos θ.
(b) csc (π − θ) = csc θ.
Exercise 40.11
Establish the following identities:
(a) sin 6x cos 2x − cos 6x sin 2x = 2 sin 2x cos 2x.
(b) sin (α + β) + sin (α − β) = 2 sin α cos β.
Exercise 40.12
Establish the following identity:
sin (α+β)
sin (α−β)
=
1+cot α tan β
.
1−cot α tan β
Exercise 40.13
Write the given expression as a function of only sin θ, cos θ, or tan θ. (k is a
given integer)
(a) cos (θ + 3π)
(b) cos [θ + (2k + 1)π]
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(c) sin (θ + 2kπ).
Exercise 40.14
Establish the identity
sin h
sin (x + h) − sin x
= cos x
+ sin x
h
h
cos h − 1
h
.
Exercise 40.15
Establish the identity
cos (x + h) − cos x
= cos x
h
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cos h − 1
h
− sin x
sin h
.
h