Find the Exact Value cos((3pi)/4-(5pi)/6)

Problem

cos((3*π)/4−(5*π)/6)

Solution

1. Identify the appropriate trigonometric identity for the cosine of a difference, which is cos(α−β)=cos(α)*cos(β)+sin(α)*sin(β)

2. Assign the values for the variables where α=(3*π)/4 and β=(5*π)/6

3. Determine the exact values of the trigonometric functions for the given angles.

cos((3*π)/4)=−√(,2)/2

sin((3*π)/4)=√(,2)/2

cos((5*π)/6)=−√(,3)/2

sin((5*π)/6)=1/2

4. Substitute these values into the difference identity.

cos((3*π)/4−(5*π)/6)=(−√(,2)/2)*(−√(,3)/2)+(√(,2)/2)*(1/2)

5. Multiply the terms in the expression.

cos((3*π)/4−(5*π)/6)=√(,6)/4+√(,2)/4

6. Combine the fractions over a common denominator.

cos((3*π)/4−(5*π)/6)=(√(,6)+√(,2))/4

Final Answer

cos((3*π)/4−(5*π)/6)=(√(,6)+√(,2))/4

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