Find the Exact Value tan(195)

Problem

tan(195)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

tan(195)=tan(150+45)

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

tan(150+45)=(tan(150)+tan(45))/(1−tan(150)*tan(45))

3. Substitute the known values for tan(150)=−√(,3)/3 and tan(45)=1

tan(195)=(−√(,3)/3+1)/(1−(−√(,3)/3)*(1))

4. Simplify the numerator and denominator by finding a common denominator of 3

tan(195)=(3−√(,3))/3/(3+√(,3))/3

5. Cancel the common denominator of 3

tan(195)=(3−√(,3))/(3+√(,3))

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

tan(195)=((3−√(,3))2)/((3+√(,3))*(3−√(,3)))

7. Expand the terms in the numerator and denominator.

tan(195)=(9−6√(,3)+3)/(9−3)

8. Combine like terms and simplify the fraction.

tan(195)=(12−6√(,3))/6

9. Divide each term in the numerator by 6

tan(195)=2−√(,3)

Final Answer

tan(195)=2−√(,3)

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