Find the Exact Value cos(60+225)

Problem

cos(60+225)

Solution

1. Identify the sum of angles formula for cosine, which is cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

2. Substitute the given angles A=60 and B=225 into the formula.

cos(60+225)=cos(60)*cos(225)−sin(60)*sin(225)

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

cos(60)=1/2

sin(60)=√(,3)/2

4. Evaluate the trigonometric functions for 225 noting it is in the third quadrant where both sine and cosine are negative.

cos(225)=−√(,2)/2

sin(225)=−√(,2)/2

5. Plug these values back into the expression.

cos(60+225)=(1/2)*(−√(,2)/2)−(√(,3)/2)*(−√(,2)/2)

6. Simplify the arithmetic by multiplying the fractions and combining terms.

cos(60+225)=−√(,2)/4+√(,6)/4

cos(60+225)=(√(,6)−√(,2))/4

Final Answer

cos(60+225)=(√(,6)−√(,2))/4

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