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9.3 Further Identities
• Double-Number Identities
– E.g. cos 2A = cos(A + A)
= cos A cos A – sin A sin A
= cos² A – sin² A
– Other forms for cos 2A are obtained by substituting either
cos² A = 1 – sin² A or sin² A = 1 – cos² A to get
cos 2A = 1 – 2 sin² A or cos 2A = 2 cos² A – 1.
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-1
9.3 Double-Number Identities
Double-Number Identities
cos 2 A cos A sin A
2
2
cos 2 A 1 2 sin A
2
sin 2 A 2 sin A cos A
2 tan A
tan 2 A
1 tan 2 A
cos 2 A 2 cos A 1
2
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-2
9.3 Finding Function Values of 2
Example Given cos 53 and sin < 0, find sin 2,
cos 2, and tan 2.
Solution To find sin 2, we must find sin .
sin 2 cos2 1
2
3
4
sin 1 sin
5
5
sin 2 2 sin cos
2
Choose the negative square
root since sin < 0.
4 3
24
sin 2 2
25
5 5
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-3
9.3 Finding Function Values of 2
cos 2 cos sin
2
2
2
2
3 4
7
25
5 5
2 tan
sin 54
4
tan 2
, where tan
3
2
1 tan
cos
3
5
2 43
24
or
2
7
1 43
sin 2 24
24
25
tan 2
7
cos 2 25 7
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-4
9.3 Simplifying Expressions Using
Double-Number Identities
Example Simplify each expression.
(a) cos² 7x – sin² 7x
(b) sin 15° cos 15°
Solution
(a) cos 2A = cos² A – sin² A. Substituting 7x in for A
gives cos² 7x – sin² 7x = cos 2(7x) = cos 14x.
(b) Apply sin 2A = 2 sin A cos A directly.
1
sin 15 cos15 (2) sin 15 cos15
2
1
1
1
sin(2 15 ) sin 30
2
2
4
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-5
9.3 Product-to-Sum Identities
• Product-to-sum identities are used in calculus to find
integrals of functions that are products of
trigonometric functions.
• Adding identities for cos(A + B) and cos(A – B)
gives
cos( A B ) cos A cos B sin A sin B
cos( A B ) cos A cos B sin A sin B
cos( A B ) cos( A B ) 2 cos A cos B
1
cos A cos B [cos( A B ) cos( A B )].
2
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-6
9.3
Product-to-Sum Identities
• Similarly, subtracting and adding the sum and
difference identities of sine and cosine, we may
derive the identities in the following table.
Product-to-Sum Identities
cos A cos B 12 [cos( A B) cos( A B)]
sin A sin B 12 [cos( A B) cos( A B)]
sin A cos B 12 [sin( A B) sin( A B)]
cos A sin B 12 [sin( A B) sin( A B)]
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-7
9.3 Using a Product-to-Sum Identity
Example Rewrite cos 2 sin as either the sum or
difference of two functions.
Solution By the identity for cos A sin A, with 2 = A
and = B,
1
cos 2 sin [sin(2 ) sin(2 )]
2
1
1
sin 3 sin .
2
2
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-8
9.3 Sum-to-Product Identities
• From the previous identities, we can derive another
group of identities that are used to rewrite sums of
trigonometric functions as products.
Sum-to-Product Identities
sin A sin B 2 sin A2 B cos A2 B
sin A sin B 2 cos A2 B sin A2 B
cos A cos B 2 cos A2 B cos A2 B
cos A cos B 2 sin A2 B sin A2 B
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-9
9.3 Using a Sum-to-Product Identity
Example Write sin 2t – sin 4t as a product of two
functions.
Solution Use the identity for sin A – sin B, with
2t = A and 4t = B.
2t 4t 2t 4t
sin 2t sin 4t 2cos
sin
2 2
6t
2t
2cos sin
2
2
2cos3t sin( t )
2cos3t sin t
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-10
9.3 Half-Number Identities
• Half-number or half-angle identities for sine and cosine are
used in calculus when eliminating the xy-term from an
equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0,
so the type of conic it represents can be determined.
• From the alternative forms of the identity for cos 2A, we can
derive three additional identities, e.g. sin A .
2
cos 2 x 1 2 sin x
2
2 sin x 1 cos 2 x
2
A
1cos 2 x
sin x
Let 2 x A so that x .
2
2
A
1cos A Choose the sign ± depending on
sin
the quadrant of the angle A/2.
2
2
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-11
9.3
Half-Number Identities
Half-Number Identities
A
1 cos A
cos
2
2
A
1 cos A
sin
2
2
A
1 cos A
A
sin A
tan
tan
2
1 cos A
2 1 cos A
A 1 cos A
tan
2
sin A
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-12
9.3 Using a Half-Number Identity to
Find an Exact Value
Example Find the exact value of cos
Solution
cos
12
.
cos 6
12
2
1 cos
6
2
3
3
1
2
1
2 3
2
2
2
22
2
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-13
9.3 Finding Function Values of x/2
Example Given cos x 23 , with 32 x 2 , find
cos 2x , sin 2x , and tan 2x .
Solution The half-angle terminates in quadrant II
since 32 x 2 34 2x .
x
1 23
1
6
sin
2
2
6 6
x
1 23
5
30
cos
2
2
6
6
6
x sin 2x
5
6
tan
30
x
2 cos 2 6
5
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-14
9.3 Simplifying Expressions Using
Half-Number Identities
1
cos
12
x
Example Simplify the expression
.
2
Solution This matches the part of the identity for
cos A/2. Replace A with 12x to get
1 cos12 x
12 x
cos
2
2
cos 6 x.
Copyright © 2011 Pearson Education, Inc.
Slide 9.3-15