unit 5.1
TRANSCRIPT
Copyright © 2011 Pearson, Inc.
5.1Fundamental
Identities
Copyright © 2011 Pearson, Inc. Slide 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 whyIdentities are important when working with trigonometric functions in calculus.
Copyright © 2011 Pearson, Inc. Slide 5.1 - 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 sincos
cot costan
Copyright © 2011 Pearson, Inc. Slide 5.1 - 4
Pythagorean Identities
2 2
2 2
2 2
cos sin 1
1 tan sec
cot 1 csc
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Example Using Identities
Find sin and cos if tan 3 and cos 0.
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Example Using Identities
To find sin, use tan 3
and cos 1 / 10.
tan sincos
sin cos tan
sin 1 / 10 3 sin 3 / 10
Find sin and cos if tan 3 and cos 0.
1 tan2 sec2 1 9 sec2
sec 10
cos 1 / 10
Therefore, cos 1 / 10 and sin 3 / 10
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Cofunction Identities
Angle A: sinAy
r tanA
y
x secA
r
x
cosAx
r cotA
x
y cscA
r
y
Angle B: sinBx
r tanB
x
y secB
r
y
cosBy
r cotB
y
x cscB
r
x
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Cofunction Identities
sin cos cos sin2 2
tan cot cot tan2 2
sec csc csc sec2 2
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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
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Example Simplifying by Factoring and Using Identities
Simplify the expression cos3 x cos xsin2 x.
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Example Simplifying by Factoring and Using Identities
cos3 x cos xsin2 x cos x(cos2 x sin2 x)
cos x(1) Pythagorean Identity
cos x
Simplify the expression cos3 x cos xsin2 x.
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Example Simplifying by Expanding and Using Identities
Simplify the expression: csc x -1 csc x 1
cos2 x
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Example Simplifying by Expanding and Using Identities
csc x 1 csc 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
cossin
1
sin2 xcsc2 x
Copyright © 2011 Pearson, Inc. Slide 5.1 - 14
Example Solving a Trigonometric Equation
Find all values of x in the interval 0,2
that solve sin3 x
cos xtan x.
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Example Solving a Trigonometric Equation
sin3 x
cos xtan x
sin3 x
cos x
sin x
cos xsin3 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
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 .
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Quick Review
Evaluate the expression.
1. sin 1 4
5
2. cos 1 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
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Quick Review Solutions
Evaluate the expression.
1. sin 1 4
5
53.13o 0.927 rad
2. cos 1 12
13
157.38o 2.747 rad
Factor the expression into a product of linear factors.
3. 2a2 3ab 2b2 2a b a 2b 4. 9u2 6u 1 3u 1 2
Simplify the expression.
5. 2
y
3
x
2x 3y
xy