Expand Using Sum/Difference Formulas cos((13pi)/12)

Problem

cos((13*π)/12)

Solution

1. Identify two angles from the unit circle that sum or subtract to (13*π)/12 We can use (9*π)/12 and (4*π)/12

2. Simplify the chosen angles to their standard forms.

(9*π)/12=(3*π)/4

(4*π)/12=π/3

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

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

4. Substitute the exact values from the unit circle for each trigonometric function.

cos((3*π)/4)=−√(,2)/2

cos(π/3)=1/2

sin((3*π)/4)=√(,2)/2

sin(π/3)=√(,3)/2

5. Multiply the terms together.

(−√(,2)/2)*(1/2)=−√(,2)/4

(√(,2)/2)*(√(,3)/2)=√(,6)/4

6. Combine the fractions to find the final result.

−√(,2)/4−√(,6)/4=(−√(,2)−√(,6))/4

Final Answer

cos((13*π)/12)=(−√(,2)−√(,6))/4

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