Simplify cos((5pi)/6-x)

Problem

cos((5*π)/6−x)

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. Substitute the values A=(5*π)/6 and B=x into the identity.

cos((5*π)/6−x)=cos((5*π)/6)*cos(x)+sin((5*π)/6)*sin(x)

3. Evaluate the trigonometric constants using the unit circle, where cos((5*π)/6)=−√(,3)/2 and sin((5*π)/6)=1/2

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

4. Simplify the expression by combining the terms over a common denominator.

cos((5*π)/6−x)=(−√(,3)*cos(x)+sin(x))/2

Final Answer

cos((5*π)/6−x)=(sin(x)−√(,3)*cos(x))/2

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