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.

sin3

cos6

cos3

sin6Express as a single trig function.

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 =

5.5.5

Finding Exact Values

2. Find the exact value for cos 150.cos 150 =

sin512

.3. Find the exact value for

5.5.6

Using the Sum and Difference Identities

Prove cos2

sin.

cos

2

sin.

sin

5.5.7

Using the Sum and Difference Identities

Given wherecos , ,

35

02

find the exact value of cos( ).

6 cos xr

5.5.8

Using the Sum and Difference Identities

xyr

A B

Given A and B

where A and B are acute angles

sin cos ,

,

2

3

4

5

find the exact value of A Bsin( ).

5.5.9

Double-Angle Identities

sin 2A = sin (A + A)

cos 2A = cos (A + A)

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:

5.5.10

Double-Angle Identities

Express each in terms of a single trig function.

a) 2 sin 0.45 cos 0.45

b) cos2 5 - sin2 5

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

5.5.11

Double-Angle Identities

Verify the identity tan A

1 cos 2 A

sin 2A.

tan A

5.5.12

Double-Angle Identities

Verify the identity tan x

sin 2x

1 cos 2x.

tan x

5.5.13

Double-Angle Equations

Find A given cos 2A =

22

where 0 A 2.

y = cos 2A2

y 2

2

5.5.14

Double-Angle Equations

Find A given sin 2A =-

12 where 0 A 2.

y = sin 2A

y 1

2

Prove

2tanx

1 tan2 x sin2x .

Identities

2sin xcos x

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:

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

The maximum distance isThe graph of sin reaches itsmaximum when

Sin 2 will reach a maximum when

5.5.18

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

B 17-24, 37-40, 43, 47, 52

5.5.19