Find the Exact Value sin((7pi)/12)

Problem

sin((7*π)/12)

Solution

1. Rewrite the angle as a sum of two known angles from the unit circle.

(7*π)/12=(3*π)/12+(4*π)/12

2. Simplify the fractions to standard angles.

(7*π)/12=π/4+π/3

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

sin(π/4+π/3)=sin(π/4)*cos(π/3)+cos(π/4)*sin(π/3)

4. Substitute the exact values for each trigonometric function.

sin(π/4)=√(,2)/2

cos(π/3)=1/2

cos(π/4)=√(,2)/2

sin(π/3)=√(,3)/2

5. Multiply the terms.

√(,2)/2⋅1/2=√(,2)/4

√(,2)/2⋅√(,3)/2=√(,6)/4

6. Combine the fractions over a common denominator.

(√(,2)+√(,6))/4

Final Answer

sin((7*π)/12)=(√(,2)+√(,6))/4

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