Question

What are all the exact solutions of -3tan^2(x)+1=0? Give your answer in radians.

Answer 1

pi/6+kpi and 5pi/6+kpi

Explanation:

Edge 2020

Answer 2

This is a problem of quadratic equation, but first of all we need to name:


`w = tan(x)`

Then, the equation above:


`-3tan^(2)(x) +1=0`

Is converting into:


`-3w^(2)+1=0`

Then:


`w= \frac{-b\pm \sqrt{b^(2)-4ac}}{2a}`

being a = -3, b = 0, c = 1, so there are two values of w:


`\left \{ {{w_(1) = ( √(3) )/(3) } \atop {w_(2) =-( √(3) )/(3) }} \right.`

Given that w = tan(x)


`\left \{ {{tan(x_(1) ) = ( √(3) )/(3) } \atop {tan(x_(2) ) =-( √(3) )/(3) }} \right.`

Finally:

`x_(1)= ( \pi )/(6)`


`x_(2)= -( \pi )/(6)`