Find the Exact Value cos(285)-cos(165)

Problem

cos(285^∘)−cos(165^∘)

Solution

1. Apply the sum-to-product formula for the difference of two cosines, which states cos(A)−cos(B)=−2*sin((A+B)/2)*sin((A−B)/2)

2. Identify the values for A and B as A=285^∘ and B=165^∘

3. Calculate the sum and difference of the angles to find the arguments for the sine functions.

(A+B)/2=(285^∘+165^∘)/2=450^∘/2=225^∘

(A−B)/2=(285^∘−165^∘)/2=120^∘/2=60^∘

4. Substitute the values back into the sum-to-product formula.

cos(285^∘)−cos(165^∘)=−2*sin(225^∘)*sin(60^∘)

5. Determine the exact values of the sine functions using the unit circle.

sin(225^∘)=−√(,2)/2

sin(60^∘)=√(,3)/2

6. Multiply the constants to find the final result.

−2⋅(−√(,2)/2)⋅√(,3)/2=√(,2)⋅√(,3)/2=√(,6)/2

Final Answer

cos(285^∘)−cos(165^∘)=√(,6)/2

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