Chapter 4Identities
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 UseFundamental identitiesEvaluating trigonometric identitiesConverting to equivalent forms
Fundamental Identities
Evaluating Trigonometric IdentitiesExample
Find the other four trigonometric functions of x when
cos x = -4/5 and tan x = 3/4
3
5
53
1
sin
1csc
5
3
4
3
5
4))(tan(cossin
3
4
431
tan
1cot
4
5
54
1
cos
1sec
xx
xxx
xx
xx
Simplifying Trigonometric Expressions
xx
x
x
x
x2
2
2
2
2
2tan
cos
sin
cos
cos11
cos
1
xx
22
tan1cos
1
xxx
xx 2cos21cottan
cottan
•Claim:
•Proof:
•Claim:
•Proof:
xxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
222
22
22
cos211
coscos1
cossin
cossin
sincos
cossin
sincos
cossin
)(cossin
)(cossin
sincos
cossin
sincos
cossin
cottan
cottan
4.2 Verifying Trigonometric Identities
Verifying identitiesTesting identities using a graphing
calculator
Verifying Identities
Verify csc(-x) = -csc x
xxxx
x cscsin
1
sin
1
)sin(
1)csc(
Verify tan x sin x + cos x = sec x
xxx
xxxx
x
xxxx sec
cos
1
cos
cossincossin
cos
sincossintan
22
Verifying Identities
Verify right-to-left:x
xxx
cos1
sincotcsc
xx
x
x
xx
x
x
xx
x
xx
xx
xx
x
x
cotcscsin
cos
sin
1
sin
cos1
sin
cos1sin
cos1
cos1sin
cos1cos1
cos1sin
cos1
sin
2
2
Verifying Identities Using a Calculator
xx
xcsc
cos1
sin2
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:
4.3 Sum, Difference, and Cofunction Identities
Sum and difference identities for cosineCofunction identitiesSum and difference identities for sine and
tangentSummary and use
Sum and Difference Identities for Cosine
cos(x – y) = cos x cos y - sin x sin y Claim: cos(/2 – y) = siny
Proof:cos(/2 – y) = cos (/2) cos y + sin(/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
yx
yx
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
yx
yxyx
tantan1
tantan
coscossinsin
coscoscoscos
coscossincos
coscoscossin
sinsincoscos
sincoscossin
cos
sintan
Finding Exact Values
Find the exact value of cos 15ºSolution:
4
132
22
13
2
1
2
1
2
3
2
1
30sin45sin30cos45cos
)3045cos(15cos
Double-Angle and Half-Angle Identities
Double-angle identitiesHalf-angle identities
Double-Angle Identities
2
2cos1cos and
2
2cos1sin 22 x
xx
x
Using Double-Angle IdentitiesExample:
Find the exact value of cos 2x if sin x = 4/5, /2 < x <
The reference angle is in the second quadrant.
25
7
5
4212cos
3
4tan,
5
4sin
31625
2
x
xx
a
Half-Angle Identities
Using a Half-Angle Identity
Example: Find cos 165º.
2
32
223
1165cos
2
330cos330cos
2
330cos1
2
330cos165cos
4.5 Product-Sum and Sum-Product Identities
Product-sum identitiesSum-product identitiesApplication
Product-Sum Identities
Using Product-Sum Identities
Example: Evaluate sin 105º sin 15º.Solution:
4
1
2
10
2
1120cos90cos
2
1
15105cos15105cos2
115sin105sin
Sum-Product Identities
Using a Sum-Product Identity
Example: Write the difference sin 7 – sin 3 as a product.
Solution:
2sin5cos22
37sin
2
37cos2
3sin7sin
Top Related