Evaluate tan165° (Use the fact that 165° = 210° - 45°)

Question

Help me evaluate tan165° using the equation

Answer 1

Recall that the sum/difference formula for tangent is as follows:

tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)*tan(b))

or

tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)*tan(b))

So,

tan(165) = tan(210 - 45) = (tan(210) - tan(45))/(1 + tan(210)*tan(45))

tan(210) is a unit circle value, which is equal to 1/√3

tan(45) is a unit circle value as well, which is equal to 1

Therefore,

(tan(210) - tan(45))/(1 + tan(210)*tan(45)) = (1/√3 - 1)/(1 + (1/√3)(1))

If you multiply the numerator and denominator by √3, you can clean things up pretty easily:

(1/√3 - 1)/(1 + (1/√3)) (√3/√3) = [(√3)(1/√3 - 1)]/[(√3)(1 + 1/√3) = (1 - √3)/(√3 + 1)

If you need to rationalize,

(1 - √3)/(√3 + 1) *(√3 - 1)/(√3 - 1) = [(1 - √3)(√3 - 1)]/[(√3 +1)(√3 - 1)] = (√3 -1 - 3 + √3)/(3 - 1) = (2√3 - 4)/2

= √3 - 2.

Hope this helps.

Mr. K