Find the Exact Value cos(255)-cos(195)

Problem

cos(255)−cos(195)

Solution

1. Identify the appropriate sum-to-product formula for the difference of two cosines.

cos(A)−cos(B)=−2*sin((A+B)/2)*sin((A−B)/2)

2. Substitute the given angles A=255 and B=195 into the formula.

cos(255)−cos(195)=−2*sin((255+195)/2)*sin((255−195)/2)

3. Simplify the arguments of the sine functions.

(255+195)/2=450/2=225

(255−195)/2=60/2=30

4. Rewrite the expression using the simplified angles.

−2*sin(225)*sin(30)

5. Evaluate the sine values using the unit circle.

sin(225)=−√(,2)/2

sin(30)=1/2

6. Multiply the values to find the final result.

−2*(−√(,2)/2)*(1/2)=√(,2)/2

Final Answer

cos(255)−cos(195)=√(,2)/2

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