Transcript 5.1 PP

5.1
Fundamental
Identities
Copyright © 2011 Pearson, Inc.
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
Basic Trigonometric Identities
Reciprocal Identities
1
cscq =
sin q
1
secq =
cosq
1
cot q =
tan q
1
sin q =
cscq
1
cosq =
secq
1
tan q =
cot q
Quotient Identities
sin q
tan q =
cosq
Copyright © 2011 Pearson, Inc.
cosq
cotq =
sin q
Slide 5.1 - 3
Pythagorean Identities
cos q + sin q = 1
2
2
1 + tan q = sec q
2
2
cot q + 1 = csc q
2
Copyright © 2011 Pearson, Inc.
2
Slide 5.1 - 4
Example Using Identities
Find sinq and cosq if tanq = 3 and cosq < 0.
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 5
Example Using Identities
Find sinq and cosq if tanq = 3 and cosq < 0.
1 + tan 2 q = sec 2 q
To find sinq , use tan q = 3
1 + 9 = sec 2 q
and cosq = -1 / 10.
secq = ± 10
cosq = -1 / 10
sin q
tan q =
cosq
sin q = cosq tan q
(
)
sin q = -1 / 10 ( 3)
sin q = -3 / 10
Therefore, cosq = -1 / 10 and sinq = -3 / 10
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 6
Cofunction Identities
y
Angle A: sin A =
r
x
cos A =
r
x
Angle B: sin B =
r
y
cos B =
r
Copyright © 2011 Pearson, Inc.
y
tan A =
x
x
cot A =
y
x
tan B =
y
y
cot B =
x
r
sec A =
x
r
csc A =
y
r
sec B =
y
r
csc B =
x
Slide 5.1 - 7
Cofunction Identities
æp
ö
sin ç - q ÷ = cos q
è2
ø
æp
ö
cos ç - q ÷ = sin q
è2
ø
æp
ö
tan ç - q ÷ = cot q
è2
ø
æp
ö
cot ç - q ÷ = tan q
è2
ø
æp
ö
sec ç - q ÷ = csc q
è2
ø
æp
ö
csc ç - q ÷ = sec q
è2
ø
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 8
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
Example Simplifying by Factoring
and Using Identities
Simplify the expression cos 3 x + cos x sin 2 x.
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 10
Example Simplifying by Factoring
and Using Identities
Simplify the expression cos 3 x + cos x sin 2 x.
cos x + cos x sin x = cos x(cos x + sin x)
3
2
2
= cos x(1)
2
Pythagorean Identity
= cos x
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 11
Example Simplifying by Expanding
and Using Identities
( csc x -1) ( csc x + 1)
Simplify the expression:
Copyright © 2011 Pearson, Inc.
cos 2 x
Slide 5.1 - 12
Example Simplifying by Expanding
and Using Identities
( csc x - 1) ( csc x + 1) = csc 2 x - 1
cos 2 x
Copyright © 2011 Pearson, Inc.
(a + b)(a - b) = a 2 - b 2
cos 2 x
cot 2 x
=
Pythagorean Identity
2
cos x
cos 2 x
1
cosq
=
×
cot q =
2
2
sin x cos x
sin q
1
=
sin 2 x
2
= csc x
Slide 5.1 - 13
Example Solving a Trigonometric
Equation
Find all values of x in the interval [ 0,2p )
sin 3 x
that solve
= tan x.
cos x
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 14
Example Solving a Trigonometric
Equation
sin 3 x
= tan x
cos x
sin 3 x sin x
=
cos x cos x
3
sin x = sin x
Reject the possibility that cos 2 x = 0
because it would make both
sides of the original equation
undefined. sin x = 0 in the interval
0 £ x < 2p when x = 0 and x = p .
sin 3 x - sin x = 0
sin x(sin 2 x - 1) = 0
(
)
2
sin
x
cos
x =0
( )
sin x = 0
or
Copyright © 2011 Pearson, Inc.
- cos 2 x = 0
Slide 5.1 - 15
Quick Review
Evaluate the expression.
æ 4ö
1. sin ç ÷
è 5ø
-1
æ 12 ö
2. cos ç - ÷
è 13 ø
-1
Factor the expression into a product of linear factors.
3. 2a 2 - 3ab - 2b 2
4. 9u - 6u + 1
2
Simplify the expression.
2 3
5. y x
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 16
Quick Review Solutions
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 17