Chapter 4 Identities - City Colleges of Chicago
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Transcript Chapter 4 Identities - City Colleges of Chicago
Chapter 4
Identities
4.1 Fundamental Identities and Their Use
4.2 Verifying Trigonometric Identities
4.3 Sum, Difference, and Cofunction Identities
4.4 Double-Angle and Half-Angle Identities
4.5 Product-Sum and Sum-Product Identities
Fundamental Identities and Their Use
Fundamental identities
Evaluating trigonometric identities
Converting to equivalent forms
Fundamental Identities
Evaluating Trigonometric Identities
Example
Find the other four trigonometric functions of x when
cos x = -4/5 and tan x = 3/4
1
1
5
1
1 4
sec x
cot x
cos x 4
4
tan x 3 3
5
4
3
4 3
sin x (cos x)(tan x)
5
5 4
1
1
5
csc x
sin x 3
3
5
Simplifying Trigonometric Expressions
•Claim:
•Proof:
•Claim:
•Proof:
1
2
1
tan
x
2
cos x
1
1 cos 2 x sin 2 x
2
1
tan
x
2
2
2
cos x
cos x
cos x
tan x cot x
1 2 cos 2 x
tan x cot x
sin x cos x
sin x cos x
tan x cot x cos x sin x sin x(cos x) cos x sin x
tan x cot x sin x cos x sin x(cos x) sin x cos x
cos x sin x
cos x sin x
sin 2 x cos 2 x 1 cos 2 x cos 2 x
2
1
2
cos
x
2
2
sin x cos x
1
4.2 Verifying Trigonometric Identities
Verifying identities
Testing identities using a graphing
calculator
Verifying Identities
Verify csc(-x) = -csc x
1
1
1
csc( x)
csc x
sin( x) sin x
sin x
Verify tan x sin x + cos x = sec x
sin x
sin 2 x cos 2 x
1
tan x sin x cos x
sin x cos x
sec x
cos x
cos x
cos x
Verifying Identities
Verify right-to-left:
sin x
csc x cot x
1 cos x
sin x
sin x 1 cos x
sin x 1 cos x
2
1 cos x 1 cos x 1 cos x
1 cos x
sin x 1 cos x 1 cos x 1 cos x csc x cot x
sin 2 x
sin x
sin x sin x
Verifying Identities Using a Calculator
Graph both sides of the equation in the same
viewing window. If they produce different graphs
they are not identities. If they appear the same
the identity must still be verified.
Example:
sin x
csc x
2
1 cos x
4.3 Sum, Difference, and Cofunction
Identities
Sum and difference identities for cosine
Cofunction identities
Sum and difference identities for sine and
tangent
Summary and use
Sum and Difference Identities for Cosine
cos(x – y) = cos x cos y - sin x sin y
Claim: cos(p/2 – y) = siny
Proof:
cos(p/2 – y) = cos (p/2) cos y + sin(p/2) sin y
= 0 cos y + 1 sin y = sin y
Sum and Difference Formula for Sine and
Tangent
sin (x- y) = sin x cos x + cos x sin y
sin x y sin x cos y cos x sin y
tan x y
cos x y cos x cos y sin x sin y
sin x cos y cos x sin y
tan x tan y
cos x cos y cos x cos y
cos x cos y sin x sin y 1 tan x tan y
cos x cos y cos x cos y
Finding Exact Values
Find the exact value of cos 15º
Solution:
cos 15 cos( 45 30)
cos 45 cos 30 sin 45 sin 30
1
3 1 1
3 1
2 2
2 2 2 2
2 3 1
4
Double-Angle and Half-Angle Identities
Double-angle identities
Half-angle identities
Double-Angle Identities
1 cos 2 x
1 cos 2 x
2
sin x
and cos x
2
2
2
Using Double-Angle Identities
Example:
Find the exact value of cos 2x if
sin x = 4/5, p/2 < x < p
The reference angle is in the
second quadrant.
a 25 16 3
4
4
sin x , tan x
5
3
2
4
7
cos 2 x 1 2
5
25
Half-Angle Identities
Using a Half-Angle Identity
Example: Find cos 165º.
cos 165 cos
330
1 cos 330
2
2
3
cos 330 cos 30
2
3
1
2 3
2
cos 165
2
2
4.5 Product-Sum and Sum-Product
Identities
Product-sum identities
Sum-product identities
Application
Product-Sum Identities
Using Product-Sum Identities
Example: Evaluate sin 105º sin 15º.
Solution:
1
sin 105 sin 15 cos105 15 cos105 15
2
1
1 1 1
cos 90 cos120 0
2
2 2 4
Sum-Product Identities
Using a Sum-Product Identity
Example: Write the difference sin 7q – sin 3q as
a product.
Solution:
sin 7q sin 3q
7q 3q
7q 3q
2 cos
sin
2
2
2 cos 5q sin 2q