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5.1
Using Fundamental Identities
Quick Review
Evaluate the expression.
4
1. sin  
5
 12 
2. cos   
 13 
Factor the expression into a product of linear factors.
-1
-1
3. 2a  3ab  2b
2
2
4. 9u  6u  1
2
Simplify the expression.
5.
2 3

y x
Quick Review Solutions
Evaluate the expression.
4
1. sin   53.13  0.927 rad
5
 12 
2. cos  
 157.38  2.747 rad
 13 
Factor the expression into a product of linear factors.
-1
-1
3. 2a  3ab  2b
2
2
4. 9u  6u  1
2
 2a  b  a  2b 
 3u  1
Simplify the expression.
5.
2 3

y x
2x  3y
xy
2
What you’ll learn about
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Identities
Basic Trigonometric Identities
Pythagorean Identities
Cofunction Identities
Odd-Even Identities
Simplifying Trigonometric Expressions
Solving Trigonometric Equations
… and why
Identities are important when working with
trigonometric functions in calculus.
Basic Trigonometric
Identities
Reciprocal Identites
1
csc  
sin 
sin  
1
csc 
1
sec  
cos 
cos  
1
sec 
Quotient Identites
sin 
tan  
cos 
cos 
cot 
tan 
1
cot  
tan 
tan  
1
cot 
Pythagorean Identities
cos   sin   1
2
2
1  tan   sec 
2
2
cot   1  csc 
2
2
Example Using Identities
Find sin  and cos if tan   3 and cos  0.
Example Using Identities
Find sin  and cos if tan   3 and cos  0.
1  tan   sec 
2
2
1  9  sec 
since tan   3
2
sec   10
cos   1/ 10
since cos   0
To find sin , use tan   3 and cos  1/ 10.
sin 
cos 
sin   cos  tan 
tan  


sin   1/ 10  3
sin   3/ 10
Therefore, cos   1/ 10 and sin   3/ 10
Cofunction Identities
y
r
x
cos A 
r
x
Angle B: sin B 
r
y
cos B 
r
Angle A:
sin A 
y
x
x
cot A 
y
x
tan B 
y
y
cot B 
x
tan A 
r
x
r
csc A 
y
r
sec B 
y
r
csc B 
x
sec A 
Cofunction Identities


sin      cos 
2



cos      sin 
2



tan      cot 
2



cot      tan 
2



sec      csc 
2



csc      sec 
2

Even-Odd Identities
sin(- x)  -sin x
cos(- x)  cos x
tan(- x)  - tan x
csc(- x)  - csc x
sec(- x)  sec x
cot(- x)  - cot x
Example Simplifying by
Factoring and Using Identities
Simplify the expression cos x  cos x sin x.
3
2
Example Simplifying by
Factoring and Using Identities
Simplify the expression cos x  cos x sin x.
3
cos x  cos x sin x
3
2
 cos x(cos x  sin x)
2
 cos x(1)
 cos x
2
Pythagorean Identity
2
Example Simplifying by
Expanding and Using Identities
Simplify the expression:
 csc x -1 csc x  1
2
cos x
Example Simplifying by
Expanding and Using Identities
Simplify the expression:
 csc x -1 csc x  1
 csc x -1 csc x  1
2
cos x
2
cos x
csc x  1

cos x
cot x

cos x
cos x
1


sin x cos x
1

sin x
 csc x
2
2
( a  b)(a - b)  a - b
2
2
2
2
2
2
2
2
2
Pythagorean Identity
cos 
cot  
sin 
Example Solving a
Trigonometric Equation
3
sin x
Find all values of x in the interval [0,2 ) that solve
 tan x.
cos x
Example Solving a
Trigonometric Equation
3
sin x
Find all values of x in the interval [0,2 ) that solve
 tan x.
cos x
3
sin x
 tan x
cos x
sin x sin x

cos x cos x
sin x  sin x
3
3
sin x  sin x  0
3
sin x(sin x  1)  0
2
 sin x  cos x   0
2
sin x  0
or
cos x  0
2
Reject the posibility that cos x  0 because it would make both
2
sides of the original equation undefined.
sin x  0 in the interval 0  x  2 when x  0 and x   .
Homework