![Page 1: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/1.jpg)
Copyright © 2011 Pearson, Inc.
5.1Fundamental
Identities
![Page 2: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/2.jpg)
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.
![Page 3: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/3.jpg)
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
![Page 4: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/4.jpg)
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
![Page 5: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/5.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 5
Example Using Identities
Find sin and cos if tan 3 and cos 0.
![Page 6: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/6.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 6
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
![Page 7: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/7.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 7
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
![Page 8: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/8.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 8
Cofunction Identities
sin cos cos sin2 2
tan cot cot tan2 2
sec csc csc sec2 2
![Page 9: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/9.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 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
![Page 10: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/10.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 10
Example Simplifying by Factoring and Using Identities
Simplify the expression cos3 x cos xsin2 x.
![Page 11: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/11.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 11
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.
![Page 12: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/12.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 12
Example Simplifying by Expanding and Using Identities
Simplify the expression: csc x -1 csc x 1
cos2 x
![Page 13: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/13.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 13
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
![Page 14: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/14.jpg)
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.
![Page 15: Copyright © 2011 Pearson, Inc. 5.1 Fundamental Identities](https://reader035.vdocuments.site/reader035/viewer/2022062805/5697bfde1a28abf838cb2352/html5/thumbnails/15.jpg)
Copyright © 2011 Pearson, Inc. Slide 5.1 - 15
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 .