Sum and Difference Formulas
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Transcript of Sum and Difference Formulas
Sum and Difference Sum and Difference FormulasFormulas
New IdentitiesNew Identities
Cosine FormulasCosine Formulas
cos cos cos sin sin
cos cos cos sin sin
Sine FormulasSine Formulas
sin sin cos cos sin
sin sin cos cos sin
Tangent FormulasTangent Formulas
tan tantan
1 tan tan
tan tantan
1 tan tan
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
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
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
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
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
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 ( )
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
Establishing an IdentityEstablishing an Identity
Establish the identityEstablish the identity
cos (cos (cos (cos (––cos cos cos cos
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
Establishing an IdentityEstablishing an Identity
Prove the identity:Prove the identity: tan (tan (= tan = tan
SolutionSolution
tan tantan( )
1 tan tantan 0
1 tan tan 0tan
tan1
Establishing an IdentityEstablishing an Identity
Prove the identity:Prove the identity:
tan cot2
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
Finding Exact Values Involving Finding Exact Values Involving Inverse Trig FunctionsInverse Trig Functions
Find the exact value of:Find the exact value of:
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)
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
SolutionSolution
12 4 5 3cos
13 5 13 5
48 15
65 65
33
65
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
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
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