Question

How do you solve `tan (x + 15°) = 3 tan x`?

Answer 1

Solve tan (x + 15) = 3tan x
Ans: x = 66.95 and 8.53 deg

Explanation:

Use trig identity: `tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)`

`tan (x + 15) = (tan x + tan 15)/(1 - tan 15.tan x)` = 3tan x.
Replace tan 15 = 0.27. Call tan x = t. We get a quadratic equation:
`- 0.80 t^2 + 2t - 0.27 = 0`
`D = d^2 = 4 - 0.86 = 3.14`--> `d = +- 1.77`

`t = tan x = 2/1.60 +- 1.77/1.60`
`t = 1.25 +- 1.10`
tan x = t = 2.15 --> `x = 66.95` deg
tan x = t = 0.15 --> `x = 8.53` deg
Check:
x = 66.95 --> tan (66.95 + 15) = tan (81.95) = 7.07
3tan (66.95) = 3(2.35) = 7.05 OK
x = 8,53 --> tan (8.53 + 15) = tan (23.53) = 0.44
3tan (8.53) = 3(0.15) = 0.45 OK