Expand Using Sum/Difference Formulas cos(165)

Problem

cos(165)

Solution

1. Identify two common angles from the unit circle that sum or subtract to 165

165=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 trigonometric values for each angle.

cos(120)=−1/2

cos(45)=√(,2)/2

sin(120)=√(,3)/2

sin(45)=√(,2)/2

4. Multiply the terms together.

(−1/2)*(√(,2)/2)−(√(,3)/2)*(√(,2)/2)

−√(,2)/4−√(,6)/4

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

(−√(,2)−√(,6))/4

Final Answer

cos(165)=(−√(,2)−√(,6))/4

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