Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities.

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.


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


Pythagorean Identities

2 2

2 2

2 2

cos sin 1

1 tan sec

cot 1 csc


Example Using Identities

Find sin and cos if tan 3 and cos 0.


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


Cofunction Identities

Angle A: sinA y

r tanA

y

x secA

r

x

cosA x

r cotA

x

y cscA

r

y

Angle B: sinB x

r tanB

x

y secB

r

y

cosB y

r cotB

y

x cscB

r

x


Cofunction Identities

sin cos cos sin2 2

tan cot cot tan2 2

sec csc csc sec2 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 cos3 x cos xsin2 x.


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.


Example Simplifying by Expanding and Using Identities

Simplify the expression: csc x -1 csc x 1

cos2 x


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 x

csc2 x


Example Solving a Trigonometric Equation

Find all values of x in the interval 0,2

that solve sin3 x

cos xtan x.


Example Solving a Trigonometric Equation

sin3 x

cos xtan x

sin3 x

cos x

sin x

cos 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

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 .

Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities.

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