Find the Exact Value cos((41pi)/12)

Problem

cos((41*π)/12)

Solution

1. Reduce the angle by subtracting multiples of 2*π to find a coterminal angle within the interval [0,2*π)

(41*π)/12−2*π=(41*π)/12−(24*π)/12=(17*π)/12

2. Rewrite the angle as a sum of two special angles whose trigonometric values are known.

(17*π)/12=(14*π)/12+(3*π)/12=(7*π)/6+π/4

3. Apply the cosine sum formula cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

cos((7*π)/6+π/4)=cos((7*π)/6)*cos(π/4)−sin((7*π)/6)*sin(π/4)

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

cos((7*π)/6)=−√(,3)/2

cos(π/4)=√(,2)/2

sin((7*π)/6)=−1/2

sin(π/4)=√(,2)/2

5. Multiply and simplify the resulting expression.

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

Final Answer

cos((41*π)/12)=(√(,2)−√(,6))/4

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