Find the Exact Value sin((13pi)/12)cos(pi/12)

Problem

sin((13*π)/12)*cos(π/12)

Solution

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

sin(A)*cos(B)=1/2*(sin(A+B)+sin(A−B))

2. Substitute the values A=(13*π)/12 and B=π/12 into the formula.

sin((13*π)/12)*cos(π/12)=1/2*(sin((13*π)/12+π/12)+sin((13*π)/12−π/12))

3. Simplify the arguments inside the sine functions.

(13*π)/12+π/12=(14*π)/12=(7*π)/6

(13*π)/12−π/12=(12*π)/12=π

4. Evaluate the sine values using the unit circle.

sin((7*π)/6)=−1/2

sin(π)=0

5. Calculate the final result by combining the values.

1/2*(−1/2+0)=−1/4

Final Answer

sin((13*π)/12)*cos(π/12)=−1/4

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