Find the Exact Value sin((5pi)/12)

Problem

sin((5*π)/12)

Solution

1. Identify the angle as a sum of two standard angles from the unit circle.

(5*π)/12=(3*π)/12+(2*π)/12

2. Simplify the fractions to express the angle in terms of π/4 and π/6

(5*π)/12=π/4+π/6

3. Apply the sum formula for sine, which is sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

sin(π/4+π/6)=sin(π/4)*cos(π/6)+cos(π/4)*sin(π/6)

4. Substitute the exact values for the trigonometric functions of the standard angles.

sin(π/4)=√(,2)/2

cos(π/6)=√(,3)/2

cos(π/4)=√(,2)/2

sin(π/6)=1/2

5. Multiply the terms together.

(√(,2)/2)*(√(,3)/2)+(√(,2)/2)*(1/2)=√(,6)/4+√(,2)/4

6. Combine the fractions over a common denominator.

(√(,6)+√(,2))/4

Final Answer

sin((5*π)/12)=(√(,6)+√(,2))/4

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