Find the Exact Value cos(135+30)

Problem

cos(135+30)

Solution

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

2. Substitute the values A=135 and B=30 into the identity.

cos(135+30)=cos(135)*cos(30)−sin(135)*sin(30)

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

cos(135)=−√(,2)/2

cos(30)=√(,3)/2

sin(135)=√(,2)/2

sin(30)=1/2

4. Plug these values back into the expression.

cos(135+30)=(−√(,2)/2)*(√(,3)/2)−(√(,2)/2)*(1/2)

5. Multiply the fractions.

cos(135+30)=−√(,6)/4−√(,2)/4

6. Combine the terms over a common denominator.

cos(135+30)=(−√(,6)−√(,2))/4

Final Answer

cos(135+30)=(−√(,6)−√(,2))/4

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