Simplify cos(285)

Problem

cos(285^∘)

Solution

1. Identify the angle in terms of a sum or difference of special angles. We can write 285^∘ as 240^∘+45^∘ or 315^∘−30^∘ 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 at these angles. Note that 240^∘ is in the third quadrant:

cos(240^∘)=−1/2

sin(240^∘)=−√(,3)/2

cos(45^∘)=√(,2)/2

sin(45^∘)=√(,2)/2

5. Plug these values back into the expression:

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

6. Simplify the arithmetic:

cos(285^∘)=−√(,2)/4+√(,6)/4

cos(285^∘)=(√(,6)−√(,2))/4

Final Answer

cos(285^∘)=(√(,6)−√(,2))/4

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