Expand Using Sum/Difference Formulas cos((17pi)/12)

Problem

cos((17*π)/12)

Solution

1. Rewrite the angle as a sum of two angles with known trigonometric values.

(17*π)/12=(9*π)/12+(8*π)/12

2. Simplify the fractions to standard angles on the unit circle.

(9*π)/12=(3*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((3*π)/4+(2*π)/3)=cos((3*π)/4)*cos((2*π)/3)−sin((3*π)/4)*sin((2*π)/3)

4. Substitute the exact values for each trigonometric function.

cos((3*π)/4)=−√(,2)/2

cos((2*π)/3)=−1/2

sin((3*π)/4)=√(,2)/2

sin((2*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((17*π)/12)=(√(,2)−√(,6))/4

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