Download - Sum and Difference Formulas
![Page 1: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/1.jpg)
Sum and Difference Sum and Difference FormulasFormulas
New IdentitiesNew Identities
![Page 2: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/2.jpg)
Cosine FormulasCosine Formulas
cos cos cos sin sin
cos cos cos sin sin
![Page 3: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/3.jpg)
Sine FormulasSine Formulas
sin sin cos cos sin
sin sin cos cos sin
![Page 4: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/4.jpg)
Tangent FormulasTangent Formulas
tan tantan
1 tan tan
tan tantan
1 tan tan
![Page 5: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/5.jpg)
Using Sum Formulas to Find Exact Using Sum Formulas to Find Exact ValuesValues
Find the exact value of cos 75Find the exact value of cos 75oo
cos 75cos 75oo = cos (30 = cos (30oo + 45 + 45oo)) cos 30cos 30oo cos 45 cos 45oo – sin 30 – sin 30oo sin 45 sin 45oo
3 2 1 2
2 2 2 2
6 2 16 2
4 4or
![Page 6: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/6.jpg)
Find the Exact ValueFind the Exact Value
Find the exact value ofFind the exact value of
7sin
12Change to degrees first (easier to find angles)
7 180105
12
sin(105 ) sin 60 45
sin 60 cos 45 sin 45 cos 60
1 2 2 3 2 6 12 6
2 2 2 2 4 4 4or
![Page 7: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/7.jpg)
Exact ValueExact Value
Find the exact value of tan 195Find the exact value of tan 195oo
tan 45 tan150tan(45 150 )
1 tan 45 tan150
1 11 1
33 31 131 1 1 13 3
3 1
3 1
![Page 8: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/8.jpg)
Using Difference Formula to Find Using Difference Formula to Find Exact ValuesExact Values
Find the exact value of Find the exact value of sin 80sin 80o o cos 20cos 20o o – sin 20– sin 20oo cos 80 cos 80oo
This is the sin difference identity so . . .This is the sin difference identity so . . .
sin(80sin(80oo – 20 – 20oo) = sin (60) = sin (60oo) = ) = 3
2
![Page 9: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/9.jpg)
Using Difference Formula to Find Using Difference Formula to Find Exact ValuesExact Values
Find the exact value ofFind the exact value of
cos 70cos 70oo cos 20 cos 20oo – sin 70 – sin 70oo sin 20 sin 20oo
This is just the cos difference formulaThis is just the cos difference formula
cos (70cos (70oo + 20 + 20oo) = cos (90) = cos (90oo) = 0) = 0
![Page 10: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/10.jpg)
Finding Exact ValuesFinding Exact Values
4If it is known that sin = , , and that
5 22 3
sin = , , find the exact value of25
a. cos ( + ) b. sin ( + ) c. tan ( )
![Page 11: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/11.jpg)
Establishing an IdentityEstablishing an Identity
Establish the Establish the identity:identity:cos( )
cot cot 1sin sin
cos cos sin sincot cot 1
sin sin
cos cos sin sincot cot 1
sin sin sin sin
cot cot 1 cot cot 1
![Page 12: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/12.jpg)
Establishing an IdentityEstablishing an Identity
Establish the identityEstablish the identity
cos (cos (cos (cos (––cos cos cos cos
![Page 13: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/13.jpg)
SolutionSolution
cos (cos (cos (cos (––cos cos cos cos cos cos cos cos ––sin sin sin sin + cos + cos cos cos sin sin
sin sin cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos = = cos cos cos cos
![Page 14: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/14.jpg)
Establishing an IdentityEstablishing an Identity
Prove the identity:Prove the identity: tan (tan (= tan = tan
![Page 15: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/15.jpg)
SolutionSolution
tan tantan( )
1 tan tantan 0
1 tan tan 0tan
tan1
![Page 16: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/16.jpg)
Establishing an IdentityEstablishing an Identity
Prove the identity:Prove the identity:
tan cot2
![Page 17: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/17.jpg)
SolutionSolution
Since tan is undefined we have to use the identity2
sin tan =
cos
sin sin cos cos sin2 2 2tan
2 cos cos sin sincos2 22
sin 0 cos 1 coscot
cos 0 sin 1 sin
![Page 18: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/18.jpg)
Finding Exact Values Involving Finding Exact Values Involving Inverse Trig FunctionsInverse Trig Functions
Find the exact value of:Find the exact value of:
![Page 19: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/19.jpg)
SolutionSolution
Think of this equation as the cos Think of this equation as the cos ((Remember that the answer Remember that the answer to an inverse trig question is an to an inverse trig question is an angle).angle).
So . . . So . . . is in the 1 is in the 1stst quadrant and quadrant and is is in the 4in the 4thth quadrant (remember range) quadrant (remember range)
![Page 20: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/20.jpg)
SolutionSolution
1 1
22 2 2 2
2 2
cos( ) cos cos sin sin
5 3tan sin
12 5
5 3tan sin
12 5
5; 12 3; 5
12 5 5 3
144 25 13 25 9 16 4
12 5 4cos ; sin cos
13 13 5
y y
x r
y x need r y r need x
r x
r x x x
![Page 21: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/21.jpg)
SolutionSolution
12 4 5 3cos
13 5 13 5
48 15
65 65
33
65
![Page 22: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/22.jpg)
Writing a Trig Expression as an Writing a Trig Expression as an Algebraic ExpressionAlgebraic Expression
Write sin (sinWrite sin (sin-1-1u + cosu + cos-1-1v) as an v) as an algebraic expression containing u algebraic expression containing u and v (without any trigonometric and v (without any trigonometric functions)functions)
Again, remember that this is just a Again, remember that this is just a sum formulasum formula
sin (sin (= sin = sin coscos + sin + sin cos cos
![Page 23: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/23.jpg)
SolutionSolution Let sinLet sin-1-1u = u = and cos and cos-1-1v = v = Then sin Then sin u and cos u and cos = v = v
2 2
2 2
1 1
2 2
cos 1 sin 1
sin 1 cos 1
sin(sin cos ) sin
sin cos cos sin
1 1
u
v
So
u v
uv u v
![Page 24: Sum and Difference Formulas](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814dc7550346895dbb1acb/html5/thumbnails/24.jpg)
On-line HelpOn-line Help TutorialsTutorials
VideosVideos