Find the Exact Value sin((43pi)/12)

Problem

sin((43*π)/12)

Solution

1. Find the coterminal angle within the standard range [0,2*π) by subtracting multiples of 2*π (which is (24*π)/12.

(43*π)/12−2*π=(43*π)/12−(24*π)/12=(19*π)/12

2. Rewrite the angle as a sum or difference of angles with known trigonometric values.

(19*π)/12=(15*π)/12+(4*π)/12=(5*π)/4+π/3

3. Apply the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

sin((5*π)/4+π/3)=sin((5*π)/4)*cos(π/3)+cos((5*π)/4)*sin(π/3)

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

sin((5*π)/4)=−√(,2)/2

cos(π/3)=1/2

cos((5*π)/4)=−√(,2)/2

sin(π/3)=√(,3)/2

5. Multiply and simplify the expression to find the final value.

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

Final Answer

sin((43*π)/12)=(−√(,2)−√(,6))/4

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