Expand Using Sum/Difference Formulas cos((7pi)/4+pi/6)

Problem

cos((7*π)/4+π/6)

Solution

1. Identify the appropriate sum formula for cosine, 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=π/6

3. Substitute these values into the formula.

cos((7*π)/4+π/6)=cos((7*π)/4)*cos(π/6)−sin((7*π)/4)*sin(π/6)

4. Determine the trigonometric values for the angles using the unit circle.

cos((7*π)/4)=√(,2)/2

sin((7*π)/4)=−√(,2)/2

cos(π/6)=√(,3)/2

sin(π/6)=1/2

5. Plug the values back into the expanded expression.

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

6. Simplify the products and combine the terms.

cos((7*π)/4+π/6)=√(,6)/4+√(,2)/4

cos((7*π)/4+π/6)=(√(,6)+√(,2))/4

Final Answer

cos((7*π)/4+π/6)=(√(,6)+√(,2))/4

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