Find the Exact Value cos((7pi)/12)

Problem

cos((7*π)/12)

Solution

1. Rewrite the angle as a sum of two known angles from the unit circle.

(7*π)/12=(3*π)/12+(4*π)/12

2. Simplify the fractions to standard angles.

(7*π)/12=π/4+π/3

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

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

4. Substitute the exact values for the sine and cosine of the standard angles.

cos(π/4)=√(,2)/2

cos(π/3)=1/2

sin(π/4)=√(,2)/2

sin(π/3)=√(,3)/2

5. Multiply the terms.

√(,2)/2⋅1/2=√(,2)/4

√(,2)/2⋅√(,3)/2=√(,6)/4

6. Subtract the results to find the final value.

√(,2)/4−√(,6)/4=(√(,2)−√(,6))/4

Final Answer

cos((7*π)/12)=(√(,2)−√(,6))/4

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