Find the Exact Value cos((7pi)/4-(11pi)/6)

Problem

cos((7*π)/4−(11*π)/6)

Solution

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

2. Assign the values for the variables where A=(7*π)/4 and B=(11*π)/6

3. Determine the exact values of the trigonometric functions for these angles using the unit circle.

4. Evaluate the cosine and sine of A

cos((7*π)/4)=√(,2)/2

sin((7*π)/4)=−√(,2)/2

5. Evaluate the cosine and sine of B

cos((11*π)/6)=√(,3)/2

sin((11*π)/6)=−1/2

6. Substitute these values into the difference identity:

cos((7*π)/4−(11*π)/6)=(√(,2)/2)*(√(,3)/2)+(−√(,2)/2)*(−1/2)

7. Simplify the expression by performing the multiplication and addition:

cos((7*π)/4−(11*π)/6)=√(,6)/4+√(,2)/4

cos((7*π)/4−(11*π)/6)=(√(,6)+√(,2))/4

Final Answer

cos((7*π)/4−(11*π)/6)=(√(,6)+√(,2))/4

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