Chapter 11: Trigonometric Identities and Equations
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Transcript Chapter 11: Trigonometric Identities and Equations
Chapter 11: Trigonometric Identities and
Equations
11.1 Trigonometric Identities
11.2 Addition and Subtraction Formulas
11.3 Double-Angle, Half-Angle, and Product-Sum
Formulas
11.4 Inverse Trigonometric Functions
11.5 Trigonometric Equations
11.3 Double-Angle, Half-Angle, and Product-Sum
Formulas
Double-Angle 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.
cos 2 A cos A sin A
2
2
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
cos 2 A 1 2 sin A
11.3 Finding Function Values of 2
Example Given cos 53 and sin < 0, find a) sin 2,
b) cos 2, and c) tan 2.
Solution To find sin 2, we must find sin .
sin 2 cos2 1
2
a) sin 2 2 sin cos
4 3
2
5 5
24
25
3
sin 1
5
9
sin 1
25
4 Choose the negative
sin
square root since sin < 0.
5
2
So cos 2 is…
8
5
94%
3%
a.
..
0%
he
3%
of
t
0%
No
ne
1.
4
2. 5
24
3. 7
7
4. 25
5. None of the above
11.3 Finding Function Values of 2
b) cos2 cos2 sin 2
2
3 4
5 5
2
7
25
2 tan
sin 54
4
c) tan2
,
where
tan
3
1 tan2
cos
3
5
2 43
1 43
2
24
OR
7
sin 2
tan 2
cos 2
24
257
25
24
7
11.3 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 cos 15 (2) sin 15 cos 15
2
1
sin( 2 15 ) 1 sin 30 1
2
2
4
11.3 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
11.3 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)]
11.3 Using a Product-to-Sum Identity
Example Rewrite cos 2 sin as the sum or difference of
two functions.
Solution By the identity for cos A sin A, with 2 = A
and = B,
cos 2 sin
1
[sin( 2 ) sin( 2 )]
2
1
1
sin 3 sin .
2
2
Since, cos A sin B 12 [sin(A B) sin(A B)]
11.3 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
11.3 Using a Sum-to-Product Identity
Example Write sin 2 – sin 4 as a product of two
functions.
Solution Use the identity for sin A – sin B, with 2 = A and
4 = B.
sin 2 sin 4
2 4 2 4 Since, sin A sin B 2 cos AB sin AB
2
2
2 cos
sin
2 2
6
2
2 cos sin
2
2
2 cos3 sin( )
2 cos3 sin
Since sine is an odd function
11.3 Half-Number Identities
• 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 2x 1 2sin 2 x
2sin 2 x 1 cos 2x
1cos 2 x
A
sin x
Let 2x A so that x .
2
2
1cos A Choose the sign ± depending on
A
sin
the quadrant of the angle A/2.
2
2
11.3 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
11.3 Using a Half-Number Identity to Find an Exact Value
Example Find the exact value of cos .
12
Solution
3
6
cos
1
2
cos
12
2
2
22
1 cos
6
2
2 3
3
2
1
2
2
11.3 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
3
3
x
x
2
2
4
2 .
2
x
1
3
a ) sin
2
2
x
1 23
b) cos
2
2
x
c) tan
2
sin
cos
x
2
x
2
1
6
6
6
5
30
6
6
6
6
30
6
5
5