Expand Using Sum/Difference Formulas cos((11pi)/12)

Problem

cos((11*π)/12)

Solution

1. Identify two common angles on the unit circle that sum or subtract to (11*π)/12 We can use (11*π)/12=(3*π)/4+π/6 because (9*π)/12+(2*π)/12=(11*π)/12

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

3. Substitute the values A=(3*π)/4 and B=π/6 into the formula.

cos((3*π)/4+π/6)=cos((3*π)/4)*cos(π/6)−sin((3*π)/4)*sin(π/6)

4. Evaluate the trigonometric functions for the specific angles.

cos((3*π)/4)=−√(,2)/2

cos(π/6)=√(,3)/2

sin((3*π)/4)=√(,2)/2

sin(π/6)=1/2

5. Multiply the terms together.

(−√(,2)/2)*(√(,3)/2)=−√(,6)/4

(√(,2)/2)*(1/2)=√(,2)/4

6. Subtract the results to find the final value.

−√(,6)/4−√(,2)/4=(−√(,6)−√(,2))/4

Final Answer

cos((11*π)/12)=(−√(,6)−√(,2))/4

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