Simplify sin(pi/12)

Problem

sin(π/12)

Solution

1. Rewrite the angle π/12 as a difference of two common angles from the unit circle.

π/12=π/4−π/6

2. Apply the sine difference formula, which states 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)

3. Substitute the known values for the sine and cosine of π/4 and π/6

sin(π/4)=√(,2)/2

cos(π/6)=√(,3)/2

cos(π/4)=√(,2)/2

sin(π/6)=1/2

4. Multiply the terms together.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

5. Subtract the resulting fractions to find the final simplified value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin(π/12)=(√(,6)−√(,2))/4

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