Find the Exact Value sin((17pi)/12)cos(pi/4)

Problem

sin((17*π)/12)*cos(π/4)

Solution

1. Identify the product-to-sum identity for sin(A)*cos(B)

sin(A)*cos(B)=1/2*(sin(A+B)+sin(A−B))

2. Substitute the values A=(17*π)/12 and B=π/4 into the identity.

sin((17*π)/12)*cos(π/4)=1/2*(sin((17*π)/12+π/4)+sin((17*π)/12−π/4))

3. Find a common denominator of 12 for the angles inside the sine functions.

π/4=(3*π)/12

4. Simplify the arguments of the sine functions.

(17*π)/12+(3*π)/12=(20*π)/12=(5*π)/3

(17*π)/12−(3*π)/12=(14*π)/12=(7*π)/6

5. Evaluate the sine values using the unit circle.

sin((5*π)/3)=−√(,3)/2

sin((7*π)/6)=−1/2

6. Combine the results and simplify the expression.

1/2*(−√(,3)/2−1/2)=1/2*((−√(,3)−1)/2)

(−√(,3)−1)/4

Final Answer

sin((17*π)/12)*cos(π/4)=(−√(,3)−1)/4

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