Find the Exact Value tan(345)

Problem

tan(345^∘)

Solution

1. Identify the angle as a sum or difference of special angles. We can write 345^∘ as 300^∘+45^∘

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=300^∘ and B=45^∘ into the formula.

4. Determine the exact values of the components: tan(300^∘)=−√(,3) and tan(45^∘)=1

5. Substitute these values into the expression:

tan(345^∘)=(−√(,3)+1)/(1−(−√(,3))*(1))

6. Simplify the expression:

tan(345^∘)=(1−√(,3))/(1+√(,3))

7. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1−√(,3)

tan(345^∘)=((1−√(,3))*(1−√(,3)))/((1+√(,3))*(1−√(,3)))

8. Expand the numerator and denominator:

tan(345^∘)=(1−2√(,3)+3)/(1−3)

9. Simplify the final fraction:

tan(345^∘)=(4−2√(,3))/(−2)

tan(345^∘)=−2+√(,3)

Final Answer

tan(345^∘)=√(,3)−2

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