Find the Exact Value cos(165)

Problem

cos(165)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

cos(165)=cos(120+45)

2. Apply the formula for the cosine of a sum, which is cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

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

3. Substitute the known exact values for the trigonometric functions of 120 and 45

cos(120)=−1/2

cos(45)=√(,2)/2

sin(120)=√(,3)/2

sin(45)=√(,2)/2

4. Multiply the terms together.

cos(165)=(−1/2)*(√(,2)/2)−(√(,3)/2)*(√(,2)/2)

cos(165)=−√(,2)/4−√(,6)/4

5. Simplify the expression by combining the fractions over a common denominator.

cos(165)=(−√(,2)−√(,6))/4

Final Answer

cos(165)=(−√(,2)−√(,6))/4

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