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
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