Find the Exact Value cos((29pi)/12)

Problem

cos((29*π)/12)

Solution

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

(29*π)/12−2*π=(29*π)/12−(24*π)/12

(29*π)/12−(24*π)/12=(5*π)/12

2. Rewrite the angle as a sum of two standard angles from the unit circle.

(5*π)/12=(3*π)/12+(2*π)/12

(5*π)/12=π/4+π/6

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

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

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

cos(π/4)=√(,2)/2

cos(π/6)=√(,3)/2

sin(π/4)=√(,2)/2

sin(π/6)=1/2

5. Simplify the expression by performing the multiplication and subtraction.

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

(√(,6)−√(,2))/4

Final Answer

cos((29*π)/12)=(√(,6)−√(,2))/4

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