Find the Exact Value cos((11pi)/12)

Problem

cos((11*π)/12)

Solution

1. Identify the angle as a sum or difference of known special angles. We can write (11*π)/12 as (3*π)/12+(8*π)/12

2. Simplify the fractions to find the standard angles.

(3*π)/12=π/4

(8*π)/12=(2*π)/3

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

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

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

cos(π/4)=√(,2)/2

cos((2*π)/3)=−1/2

sin(π/4)=√(,2)/2

sin((2*π)/3)=√(,3)/2

5. Multiply the terms together.

(√(,2)/2)*(−1/2)−(√(,2)/2)*(√(,3)/2)

−√(,2)/4−√(,6)/4

6. Combine the fractions over a common denominator.

(−√(,2)−√(,6))/4

Final Answer

cos((11*π)/12)=(−√(,2)−√(,6))/4

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