Find the Exact Value sin((23pi)/12)

Problem

sin((23*π)/12)

Solution

1. Identify the angle in terms of a reference angle or a sum/difference of known angles. Note that (23*π)/12 is in the fourth quadrant and can be written as 2*π−π/12

2. Apply the periodicity and symmetry properties of the sine function.

sin(2*π−π/12)=−sin(π/12)

3. Rewrite the angle π/12 as a difference of two standard angles, such as π/4−π/6

−sin(π/4−π/6)

4. Apply the sine difference formula sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

−(sin(π/4)*cos(π/6)−cos(π/4)*sin(π/6))

5. Substitute the exact values for the trigonometric functions: sin(π/4)=√(,2)/2 cos(π/6)=√(,3)/2 cos(π/4)=√(,2)/2 and sin(π/6)=1/2

−(√(,2)/2⋅√(,3)/2−√(,2)/2⋅1/2)

6. Simplify the expression by multiplying the fractions and distributing the negative sign.

−(√(,6)/4−√(,2)/4)

7. Combine the terms into a single fraction.

(√(,2)−√(,6))/4

Final Answer

sin((23*π)/12)=(√(,2)−√(,6))/4

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