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Unit 5.1
- 2. What you’ll learn about
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.
Copyright © 2011 Pearson, Inc. Slide 5.1 - 2
- 3. Basic Trigonometric Identities
Reciprocal Identites
csc
1
sin
sec
1
cos
cot
1
tan
sin
1
csc
cos
1
sec
tan
1
cot
Quotient Identites
tan
sin
cos
cot
cos
tan
Copyright © 2011 Pearson, Inc. Slide 5.1 - 3
- 4. Pythagorean Identities
2 2
cos sin 1
1 tan sec
cot 1 csc
2 2
2 2
Copyright © 2011 Pearson, Inc. Slide 5.1 - 4
- 6. Example Using Identities
Find sin and cos if tan 3 and cos 0.
To find sin , use tan 3
and cos 1 / 10.
tan
sin
cos
sin cos tan
sin 1 / 103
sin 3 / 10
1 tan2 sec2
1 9 sec2
sec 10
cos 1 / 10
Therefore, cos 1/ 10 and sin 3/ 10
Copyright © 2011 Pearson, Inc. Slide 5.1 - 6
- 7. Cofunction Identities
Angle A: sin A
y
r
tan A
y
x
secA
r
x
cosA
x
r
cot A
x
y
cscA
r
y
Angle B: sin B
x
r
tan B
x
y
secB
r
y
cosB
y
r
cot B
y
x
cscB
r
x
Copyright © 2011 Pearson, Inc. Slide 5.1 - 7
- 8. Cofunction Identities
sin cos cos sin
2 2
tan cot cot tan
2 2
sec csc csc sec
2 2
Copyright © 2011 Pearson, Inc. Slide 5.1 - 8
- 9. 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
Copyright © 2011 Pearson, Inc. Slide 5.1 - 9
- 10. Example Simplifying by Factoring
and Using Identities
Simplify the expression cos3 x cos x sin2 x.
Copyright © 2011 Pearson, Inc. Slide 5.1 - 10
- 11. Example Simplifying by Factoring
and Using Identities
Simplify the expression cos3 x cos x sin2 x.
cos3 x cos xsin2 x cos x(cos2 x sin2 x)
cos x(1) Pythagorean Identity
cos x
Copyright © 2011 Pearson, Inc. Slide 5.1 - 11
- 12. Example Simplifying by Expanding
and Using Identities
Simplify the expression:
csc x -1csc x 1
cos2 x
Copyright © 2011 Pearson, Inc. Slide 5.1 - 12
- 13. Example Simplifying by Expanding
and Using Identities
csc x 1csc x 1
cos2 x
csc2 x 1
cos2 x
(a b)(a b) a2 b2
cot2 x
cos2 x
Pythagorean Identity
cos2 x
sin2 x
1
cos2 x
cot
cos
sin
1
sin2 x
csc2 x
Copyright © 2011 Pearson, Inc. Slide 5.1 - 13
- 14. Example Solving a Trigonometric
Equation
Find all values of x in the interval 0,2
that solve
sin3 x
cos x
tan x.
Copyright © 2011 Pearson, Inc. Slide 5.1 - 14
- 15. Example Solving a Trigonometric
Equation
sin3 x
cos x
tan x
sin3 x
cos x
sin x
cos x
Reject the posibility that cos2 x 0
because it would make both
sides of the original equation
undefined. sin x 0 in the interval
0 x 2 when x 0 and x .
sin3 x sin x
sin3 x sin x 0
sin x(sin2 x 1) 0
sin x cos2 x 0
sin x 0 or cos2 x 0
Copyright © 2011 Pearson, Inc. Slide 5.1 - 15
- 16. Quick Review
Evaluate the expression.
1. sin1 4
5
2. cos1
12
13
Factor the expression into a product of linear factors.
3. 2a2 3ab 2b2
4. 9u2 6u 1
Simplify the expression.
5.
2
y
3
x
Copyright © 2011 Pearson, Inc. Slide 5.1 - 16
- 17. Quick Review Solutions
Evaluate the expression.
1. sin1 4
5
53.13o 0.927 rad
2. cos1
12
13
157.38o 2.747 rad
Factor the expression into a product of linear factors.
3. 2a2 3ab 2b2 2a ba 2b
4. 9u2 6u 1 3u 12
Simplify the expression.
5.
2
y
3
x
2x 3y
xy
Copyright © 2011 Pearson, Inc. Slide 5.1 - 17