Hprec9 3
-
Upload
stevenhbills -
Category
Business
-
view
135 -
download
0
description
Transcript of Hprec9 3
9-3: Other IdentitiesObjectives:•Use the double angle and half angle identities for sine, cosine & tangent
©2009 Roy L. Gover (www.mrgover.com)
)
Analysis
Does sin 90º =2 sin 45°?
Does cos 30°= cos 60°?
1
2
Analysis
sin( )
sin cos cos sin
From Lesson 9-2:
Let , then… x
Important Idea
sin 2 2sin cosx x x
Double Angle Identity for sine:
Example
If sin x = 4/5 and x is in the first quadrant, find the exact value of sin 2x.…first, find cos x…
Try This
If sin x = 3/5 and x is in the first quadrant, find the exact value of sin 2x
24sin 2
25x
Try ThisIf sin x = 3/5 and x is in the second quadrant, find the exact value of sin 2x. 24
sin 225
x
Analysis
cos( )
cos cos sin sin
From Lesson 9-2:
Let , then… x
Important Idea
2 2cos 2 cos sin x x x
Double Angle Identity for cosine:
Double Angle Identity for cos
Three forms...
1. cos 2x = cos2x - sin2x
2. cos 2x = 2cos2x - 1
3. cos 2x = 1 - 2sin2x
Important Idea
Example
If sin x is 1/5 and x is in the first quadrant, find the exact value of cos 2x.
Try ThisIf sin x is 2/5 and x is in the second quadrant, find the exact value of cos 2x.
17cos 2
25x
What would you do to derive a double angle identity for tangent?
Double Angle Identity for Tangent:
2
2 tantan 2
1 tan
x
xx
Important Idea
Try This
If , and is in the first quadrant, find .
5cos
5x x
tan 2x
Hint: first find …
tan x
4tan 2
3x
Analysis
2cos 2 2cos 1
1 cos 2cos
2
One of the double angle identities:
Solve for cos :
Analysis
1 cos 2os
2c
Let 22
x
x
Analysis
1 coscos
2 2
x x
which results in:
The half angle identity for cosine.
Important Idea
1 cossin
2 2
x x
1 coscos
2 2
x x
1 costan
2 1 cos
x x
x
The half angle identities: See
page 596 of your text.
Example Use a half angle identity to find the exact value of sin 22°30’Hint: Multiply the given angle by 2 then divide by 2
Example Use a half angle identity to find the exact value of19
cos12
Hint: Multiply the given angle by 2 then divide by 2
Try This Use a half angle identity to find the exact value of cos 105°
210 2 3cos105 cos
2 2
Try This
Use a half angle identity
to find the exact value of
55 2 24sin sin8 2 2
5sin
8