Find the Exact Value tan(285)

Problem

tan(285^∘)

Solution

1. Identify the angle as a sum or difference of special angles. We can write 285^∘ as 225^∘+60^∘

2. Apply the formula for the tangent of a sum: tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

3. Substitute the values A=225^∘ and B=60^∘ into the formula.

4. Evaluate the trigonometric functions: tan(225^∘)=1 and tan(60^∘)=√(,3)

5. Simplify the expression:

tan(285^∘)=(1+√(,3))/(1−(1)*(√(,3)))

tan(285^∘)=(1+√(,3))/(1−√(,3))

6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1+√(,3)

tan(285^∘)=((1+√(,3))*(1+√(,3)))/((1−√(,3))*(1+√(,3)))

tan(285^∘)=(1+2√(,3)+3)/(1−3)

tan(285^∘)=(4+2√(,3))/(−2)

7. Divide each term in the numerator by −2

tan(285^∘)=−2−√(,3)

Final Answer

tan(285^∘)=−2−√(,3)

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