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Arkansas Tech University
MATH 1203: Trigonometry
Dr. Marcel B. Finan
21
Conversion Identities
In this section, you will learn (1) how to restate a product of two trigonometric functions as a sum, (2) how to restate a sum of two trigonometric
functions as a product, and (3) how to write a sum of two trigonometric
functions as a single function.
Product-To-Sum Identities
By the addition and subtraction formulas for the cosine, we have
cos (x + y) = cos x cos y − sin x sin y
(1)
cos (x − y) = cos x cos y + sin x sin y.
(2)
and
Adding these equations together to obtain
2 cos x cos y = cos (x + y) + cos (x − y)
(3)
1
cos x cos y = [cos (x + y) + cos (x − y)]
2
(4)
or
Subtracting( 1) from ( 2) to obtain
2 sin x sin y = cos (x − y) − cos (x + y)
(5)
1
sin x sin y = [cos (x − y) − cos (x + y)].
2
(6)
or
Now, by the addition and subtraction formulas for the sine, we have
sin (x + y) = sin x cos y + cos x sin y
sin (x − y) = sin x cos y − cos x sin y.
1
Adding these equations together to obtain
2 sin x cos y = sin (x + y) + sin (x − y)
(7)
1
sin x cos y = [sin (x + y) + sin (x − y)].
2
(8)
or
Identities ( 4), ( 6), and ( 8) are known as the product-to-sum identities.
Example 21.1
Write sin 3x cos x as a sum/difference containing only sines and cosines.
Solution.
Using ( 8) we obtain
1
sin 3x cos x = [sin (3x + x) + sin (3x − x)]
2
1
= (sin 4x + sin 2x)
2
Sum-to-Product Identities
We next derive the so-called sum-to-product identities. For this purpose,
we let α = x + y and β = x − y. Solving for x and y in terms of α and β we
find
α+β
α−β
x=
and y =
.
2
2
By identity ( 3) we find
cos α + cos β = 2 cos (
α+β
α−β
) cos (
).
2
2
(9)
α+β
α−β
) sin (
).
2
2
(10)
Using identity ( 5) we find
cos α − cos β = −2 sin (
Now, by identity ( 7) we have
sin α + sin β = 2 sin (
α−β
α+β
) cos (
).
2
2
2
(11)
Using this last identity by replacing β by −β and using the fact that the sine
function is odd we find
sin α − sin β = 2 sin (
α+β
α−β
) cos (
).
2
2
(12)
Formulas (9) - (12) are known as the sum-to-product formulas.
Example 21.2
Establish the identity:
cos 2x+cos 2y
cos 2x−cos 2y
= − cot (x + y) cot (x − y).
Solution.
Using the product-to-sum identities we find
2 cos ( 2x+2y
) cos ( 2x−2y
)
cos 2x + cos 2y
2
2
=
2x+2y
2x−2y
cos 2x − cos 2y −2 sin ( 2 ) sin ( 2 )
= − cot (x + y) cot (x − y)
Writing a sin x + b cos x in the Form k sin (x + θ).
Let P (a, b) be a coordinate point in the plane and let θ be the angle with
−→
initial side the x-axis and terminal side the ray OP as shown in Figure 21.1
Figure 21.1
Let k =
√
a2 + b2 . Then, according to Figure 91 we have
a
b
cos θ = √
and sin θ = √
.
2
2
2
a +b
a + b2
3
Then in terms of k and θ we can write
√
a sin x + b cos x = a2 + b2
a
b
√
sin x + √
cos x
a2 + b 2
a2 + b 2
=k(cos θ sin x + sin θ cos x) = k sin (x + θ).
Example 21.3
Write y = 21 sin x − 12 cos x in the form y = k sin (x + θ).
Solution.
Since a =
√
2
, sin θ
2
=
1
2
b
k
and b =
√
=−
2
.
2
− 12
q
we find k = ( 12 )2 + (− 21 )2 =
Thus θ = −45◦ and
√
2
y=
sin (x − 45◦ ).
2
4
√
2
, cos θ
2
=
a
k
=