sum and difference formulas new identities. cosine 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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