# Sum and Difference Formulas New Identities. Cosine Formulas.

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Sum and Difference Formulas New Identities

Cosine Formulas

Sine Formulas

Tangent Formulas

Using Sum Formulas to Find Exact Values Find the exact value of cos 75 o Find the exact value of cos 75 o cos 75 o = cos (30 o + 45 o ) cos 75 o = cos (30 o + 45 o ) cos 30 o cos 45 o – sin 30 o sin 45 o cos 30 o cos 45 o – sin 30 o sin 45 o

Find the Exact Value Find the exact value of Find the exact value of

Exact Value Find the exact value of tan 195 o Find the exact value of tan 195 o

Using Difference Formula to Find Exact Values Find the exact value of Find the exact value of sin 80 o cos 20 o – sin 20 o cos 80 o sin 80 o cos 20 o – sin 20 o cos 80 o This is the sin difference identity so... This is the sin difference identity so... sin(80 o – 20 o ) = sin (60 o ) = sin(80 o – 20 o ) = sin (60 o ) =

Using Difference Formula to Find Exact Values Find the exact value of Find the exact value of cos 70 o cos 20 o – sin 70 o sin 20 o cos 70 o cos 20 o – sin 70 o sin 20 o This is just the cos difference formula This is just the cos difference formula cos (70 o + 20 o ) = cos (90 o ) = 0 cos (70 o + 20 o ) = cos (90 o ) = 0

Finding Exact Values

Establishing an Identity Establish the identity: Establish the identity:

Establishing an Identity Establish the identity Establish the identity cos (  cos (  –  cos  cos  cos (  cos (  –  cos  cos 

Solution cos  cos  –  sin  sin  + cos  cos  sin  sin  cos  cos  –  sin  sin  + cos  cos  sin  sin  cos  cos  cos  cos  cos  cos  cos  cos  cos  cos  cos  cos   cos  cos  =  cos  cos   cos  cos  =  cos  cos 

Establishing an Identity Prove the identity: Prove the identity: tan (  = tan  tan (  = tan 

Solution

Establishing an Identity Prove the identity: Prove the identity:

Solution

Finding Exact Values Involving Inverse Trig Functions Find the exact value of: Find the exact value of:

Solution Think of this equation as the cos (   Remember that the answer to an inverse trig question is an angle). Think of this equation as the cos (   Remember that the answer to an inverse trig question is an angle). So...  is in the 1 st quadrant and  is in the 4 th quadrant (remember range) So...  is in the 1 st quadrant and  is in the 4 th quadrant (remember range)

Solution

Solution

Writing a Trig Expression as an Algebraic Expression Write sin (sin -1 u + cos -1 v) as an algebraic expression containing u and v (without any trigonometric functions) Write sin (sin -1 u + cos -1 v) as an algebraic expression containing u and v (without any trigonometric functions) Again, remember that this is just a sum formula Again, remember that this is just a sum formula sin (  = sin  cos  + sin  cos  sin (  = sin  cos  + sin  cos 

Solution Let sin -1 u =  and cos -1 v =  Let sin -1 u =  and cos -1 v =  Then sin  u and cos  = v Then sin  u and cos  = v

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