Expand Using Sum/Difference Formulas sin((11pi)/12)

Problem

sin((11*π)/12)

Solution

1. Identify two common angles on the unit circle that sum or subtract to (11*π)/12 We can use (11*π)/12=(8*π)/12+(3*π)/12

2. Simplify the fractions to standard angles.

(8*π)/12=(2*π)/3

(3*π)/12=π/4

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

sin((2*π)/3+π/4)=sin((2*π)/3)*cos(π/4)+cos((2*π)/3)*sin(π/4)

4. Substitute the exact values from the unit circle.

sin((2*π)/3)=√(,3)/2

cos(π/4)=√(,2)/2

cos((2*π)/3)=−1/2

sin(π/4)=√(,2)/2

5. Multiply the terms.

(√(,3)/2)*(√(,2)/2)=√(,6)/4

(−1/2)*(√(,2)/2)=−√(,2)/4

6. Combine the fractions over a common denominator.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin((11*π)/12)=(√(,6)−√(,2))/4

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