Find the Exact Value cos(285)

Problem

cos(285)

Solution

1. Identify the angle as a sum or difference of special angles. We can write 285 as 225+60 or 240+45 Let's use 240+45

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

3. Substitute the values A=240 and B=45 into the formula:

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

4. Determine the exact values for the trigonometric functions of the special angles.

cos(240)=−1/2

sin(240)=−√(,3)/2

cos(45)=√(,2)/2

sin(45)=√(,2)/2

5. Substitute these values back into the expression:

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

6. Simplify the products:

cos(285)=−√(,2)/4+√(,6)/4

7. Combine the terms over a common denominator:

cos(285)=(√(,6)−√(,2))/4

Final Answer

cos(285)=(√(,6)−√(,2))/4

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