using sum, difference, and double-angle identities

MATHPOWERTM 12, WESTERN EDITION

5.5

5.5.1

Chapter 5 Trigonometric Equations

5.5.2

Sum and Difference Identities

sin(A + B) = sin A cos B + cos A sin B

sin(A - B) = sin A cos B - cos A sin B

cos(A + B) = cos A cos B - sin A sin B

cos(A - B) = cos A cos B + sin A sin B

tan(A B) tanA tanB

1 tan Atan B

tan(A B) tanA tanB

1 tan Atan B

5.5.3

Simplifying Trigonometric Expressions

Express cos 1000 cos 800 + sin 800 sin 1000 as a trig function of a single angle.

This function has the same pattern as cos (A - B),with A = 1000 and B = 800.

cos 100 cos 80 + sin 80 sin 100 = cos(1000 - 800) = cos 200

sin3

cos6

cos3

sin6Express as a single trig function.

This function has the same pattern as sin(A - B), with A

3

and B 6

.

sin3

cos6

cos3

sin6

sin3

6

sin6

1.

2.

5.5.4

Finding Exact Values

1. Find the exact value for sin 750.Think of the angle measures that produce exact values:300, 450, and 600.Use the sum and difference identities. Which angles, used in combination of addition or subtraction, would give a result of 750?

sin 750 = sin(300 + 450) = sin 300 cos 450 + cos 300 sin 450

1

2

2

2

3

2

2

2

2 6

4

5.5.5

Finding Exact Values

2. Find the exact value for cos 150.cos 150 = cos(450 - 300) = cos 450 cos 300 + sin 450 sin300

2

2

3

2

2

2

1

2

6 2

4

sin512

.3. Find the exact value for 3

412

4

312

6

212

sin512

sin(4

6

)

sin4

cos6

cos4

sin6

2

2

3

2

2

2

1

2

6 2

4

5.5.6

Using the Sum and Difference Identities

Prove cos2

sin.

cos

2

sin.

cos2

cos sin2

sin

(0)(cos) (1)(sin)

(1)(sin)

sin

sin

L.S. = R.S.

5.5.7


x = 3r = 5

r2 = x2 + y2

y2 = r2 - x2 = 52 - 32

= 16 y = ± 4

Given wherecos , ,

35

02

find the exact value of cos( ).

6

cos( ) cos cos sin sin

6 6 6

cos xr

( )( ) ( )( )35

32

45

12

3 3 4

10

Therefore, cos( ) .

6

3 3 410

5.5.8


xyr

A B

23

4

53

Given A and B

where A and B are acute angles

sin cos ,

,

2

3

4

5

find the exact value of A Bsin( ).

( )( ) ( )( )23

45

53

35

8 3 515

Therefore A B, sin( ) . 8 3 515

5sin( ) sin cos cos sinA B A B A B

5.5.9

Double-Angle Identities

sin 2A = sin (A + A)

= sin A cos A + cos A sin A

= 2 sin A cos A

cos 2A = cos (A + A)

= cos A cos A - sin A sin A

= cos2 A - sin2A

The identities for the sine and cosine of the sum of twonumbers can be used, when the two numbers A and Bare equal, to develop the identities for sin 2A and cos 2A.

Identities for sin 2x and cos 2x:

sin 2x = 2sin x cos x cos 2x = cos2x - sin2xcos 2x = 2cos2x - 1cos 2x = 1 - 2sin2x

5.5.10


Express each in terms of a single trig function.

a) 2 sin 0.45 cos 0.45

sin 2x = 2sin x cos xsin 2(0.45) = 2sin 0.45 cos 0.45 sin 0.9 = 2sin 0.45 cos 0.45

b) cos2 5 - sin2 5

cos 2x = cos2 x - sin2 x cos 2(5) = cos2 5 - sin2 5 cos 10 = cos2 5 - sin2 5

Find the value of cos 2x for x = 0.69.

cos 2x = cos2 x - sin2 xcos 2(0.69) = cos2 0.69 - sin2 0.69 cos 2x = 0.1896

A group of Friars opened a florist shop to help with their belfry payments. Everyone liked to buy flowers from the Men of God, so their business flourished. A rival florist became upset that his business was suffering because people felt compelled to buy from the Friars, so he asked the Friars to cut back hours or close down. The Friars refused. The florist went to them and begged that they shut down.

5.5.11


Verify the identity tan A

1 cos 2 A

sin 2A.

1 (cos2 A sin2 A)

2sin Acos A

1 cos2 A sin2 A)

2 sinAcos A

sin2 A sin2 A

2sin Acos A

2sin 2 A

2sin Acos A

sin A

cos A

tan A

tan A

L.S = R.S.

5.5.12


Verify the identity tan x

sin 2x

1 cos 2x.

2sin x cos x1 2 cos 2 x 1

2sin x cos x2cos2 x

sin xcos x

tan x

tan x

L.S = R.S.

5.5.13

Double-Angle Equations

Find A given cos 2A =

22

where 0 A 2.

y = cos 2A

2 A

A 8

,78

,98

,15

8

2

8

78

98

158 y

2

2

For cos A = 22

where 0 A 2, A

4

and 74

.

2 A 4

2n,74

2n.4

,74

,94

,15

4

5.5.14

Double-Angle Equations

Find A given sin 2A =-

12 where 0 A 2.

76

,

A 712

,1112

,1912

,2312

116

,19

6,

236

y = sin 2A

y 1

2712

1112

1912

2312

2 A

For sin A =-

12 where 0 A 2, A=

76

and11

6.

2 A 76

2n,11

6 2n.

Graph the function f (x) 2 tanx

1 tan2 x

Using Technology

and predict the period.

The period is

Rewrite f(x) as a single trig function: f(x) = sin 2x

5.5.15

Prove

2tanx

1 tan2 x sin2x .

Identities

2sin x

cos xsec 2 x

2sin x

cos x1

cos 2 x

2sin x

cos x

cos 2 x

1

2sin xcos x

2sin xcos x

L.S. = R.S.5.5.16

Applying Skills to Solve a Problem

The horizontal distance that a soccer ball will travel, when

kicked at an angle , is given by d 2v0

2

gsincos ,

where d is the horizontal distance in metres, v0 is the initialvelocity in metres per second, and g is the acceleration dueto gravity, which is 9.81 m/s2.

a) Rewrite the expression as a sine function.Use the identity sin 2A = 2sin A cos A:

d 2v0

2

gsincos

d v0

2

gsin2

5.5.17

b) Find the distance when the initial velocity is 20 m/s.

Applying Skills to Solve a Problem [cont’d]

d v0

2

gsin2

d (20)2

9.81sin2

Dis

tan

ce

Angle

From the graph, the maximumdistance occurs when = 0.785398.

The maximum distance is 40.7747 m.The graph of sin reaches itsmaximum when

2

.

Sin 2 will reach a maximum when

2

2

or 4

.

5.5.18

Suggested Questions:Pages 272-274A 1-16, 25-35 odd

B 17-24, 37-40, 43, 47, 48

5.5.19

Again they refused. So the florist then hired Hugh McTaggert, the biggest meanest thug in town. He went to the Friars' shop beat them up, destroyed their flowers, trashed their shop, and said that if they didn't close, he'd be back. Well, totally terrified, the Friars closed up shop and hid in their rooms. This proved that Hugh, and only Hugh, can prevent florist friars.

using sum, difference, and double-angle identities

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