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

Problem

cos(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 A=π/4 and B=(11*π)/6 to the identity.

3. Substitute the values into the formula:

cos(π/4−(11*π)/6)=cos(π/4)*cos((11*π)/6)+sin(π/4)*sin((11*π)/6)

4. Determine the exact values of the trigonometric functions for the given angles using the unit circle:

cos(π/4)=√(,2)/2

sin(π/4)=√(,2)/2

cos((11*π)/6)=√(,3)/2

sin((11*π)/6)=−1/2

5. Multiply the terms together:

(√(,2)/2)*(√(,3)/2)+(√(,2)/2)*(−1/2)

6. Simplify the expression by performing the multiplication and combining the fractions:

√(,6)/4−√(,2)/4

(√(,6)−√(,2))/4

Final Answer

cos(π/4−(11*π)/6)=(√(,6)−√(,2))/4

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