Expand Using Sum/Difference Formulas cos(285)

Problem

cos(285)

Solution

1. Identify two angles that sum or subtract to 285 using standard angles from the unit circle.

285=240+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(240+45)=cos(240)*cos(45)−sin(240)*sin(45)

3. Determine the exact values for the trigonometric functions at the specified angles.

cos(240)=−1/2

sin(240)=−√(,3)/2

cos(45)=√(,2)/2

sin(45)=√(,2)/2

4. Substitute these values into the expanded expression.

cos(285)=(−1/2)*(√(,2)/2)−(−√(,3)/2)*(√(,2)/2)

5. Simplify the products and combine the terms.

cos(285)=−√(,2)/4+√(,6)/4

cos(285)=(√(,6)−√(,2))/4

Final Answer

cos(285)=(√(,6)−√(,2))/4

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