chapter 5 trigonometric...

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

sin

3cos

6 cos

3sin

6Express as a single trig function.

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

A

3 and B

6.

sin

3cos

6 cos

3sin

6 sin

3

6

sin

6

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

s in5

12.3. Find the exact value for

3

4

12

4

3

12

6

2

12

sin5

12 sin(

4

6)

s in

4cos

6 cos

4sin

6

2

2

3

2

2

2

1

2

6 2

4

5.5.6

Using the Sum and Difference Identities

Prove cos

2

sin.

cos

2

sin.

cos

2cos sin

2sin

(0)(cos) (1)(s in)

(1)(s in)

sin

sin

L.S. = R.S.

5.5.7


x = 3

r = 5

r2 = x2 + y2

y2 = r2 - x2

= 52 - 32

= 16

y = ± 4

Given wherecos , ,

3

50

2

find the exact value of cos( ).

6

cos( ) cos cos sin sin

6 6 6

cos x

r

( )( ) ( )( )3

5

3

2

4

5

1

2

3 3 4

10

Therefore, cos( ) .

6

3 3 4

10

5.5.8


x

y

r

A B

2

3

4

5

3

Given A and B

where A and B are acute angles

sin cos ,

,

2

3

4

5

find the exact value of A Bsin( ).

( )( ) ( )( )2

3

4

5

5

3

3

5

8 3 5

15

Therefore A B, sin( ) . 8 3 5

15

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 two

numbers can be used, when the two numbers A and B

are 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 - sin2x

cos 2x = 2cos2x - 1

cos 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 x

sin 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 x

cos 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)

2sinAcosA

1 cos2 A sin2 A)

2 sinAcosA

sin2 A sin2 A

2sinAcosA

2sin2 A

2sinAcosA

s inA

cosA

tan A

tan A

L.S = R.S.

5.5.12


Verify the identity tan x

sin 2x

1 cos 2x.

2sin x cos x

1 2 cos 2

x 1

2sin x cos x

2cos2

x

s in x

cos x

tan x

tan x

L.S = R.S.

5.5.13

Double-Angle Equations

Find A given cos 2A =

2

2 where 0 A 2.

y = cos 2A

2A

A

8,7

8,9

8,15

8

2

8

7

8

9

8

15

8 y 2

2

For cos A = 2

2 where 0 A 2, A

4 and

7

4.

2 A

4 2n,

7

4 2n.

4,

7

4,

9

4,

15

4

5.5.14

Double-Angle Equations

Find A given sin 2A = -

1

2 where 0 A 2.

7

6,

A 7

12,11

12,19

12,23

12

11

6,

19

6,

23

6

y = sin 2A

y 1

27

12

11

12

19

12

23

12

2A

For sin A = -

1

2 where 0 A 2, A =

7

6 and

11

6.

2 A7

6 2n,

11

6 2n.

Graph the function f (x) 2 tanx

1 tan2x

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 s in2x.

Identities

2sinx

cosx

sec 2

x

2s inx

cosx1

cos 2

x

2sinx

cosx

cos 2 x

1

2sinxcosx

2sinxcosx

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 initial

velocity in metres per second, and g is the acceleration due

to 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

gs in2

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

gs in2

d (20)2

9.81sin2

Dis

tan

ce

Angle

From the graph, the maximum

distance 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 272-274

A 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.

chapter 5 trigonometric...

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