Simplify cos(120+45)

Problem

cos(120^∘+45^∘)

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=120^∘ and B=45^∘ into the identity.

cos(120^∘+45^∘)=cos(120^∘)*cos(45^∘)−sin(120^∘)*sin(45^∘)

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

cos(120^∘)=−1/2

sin(120^∘)=√(,3)/2

cos(45^∘)=√(,2)/2

sin(45^∘)=√(,2)/2

4. Plug these values back into the expression.

cos(120^∘+45^∘)=(−1/2)*(√(,2)/2)−(√(,3)/2)*(√(,2)/2)

5. Multiply the fractions.

cos(120^∘+45^∘)=−√(,2)/4−√(,6)/4

6. Combine the terms over a common denominator.

cos(120^∘+45^∘)=(−√(,2)−√(,6))/4

Final Answer

cos(120^∘+45^∘)=(−√(,2)−√(,6))/4

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