Question

Use a calculator to solve the equation on the interval [0 , 2π].
2 tan²(x) - 7 tan(x) + 5 =0

Answer 1

`\theta_(1) \approx 0.379\cdot \pi` or
`\theta_(1) \approx 1.379\cdot \pi`,
`\theta_(2) = 0.25\cdot \pi` or
`\theta_(2) = 1.25\cdot \pi`

Explanation:

Let use the following substitution formula:

`u = \tan \theta`

The trigonometric expression is converted into an algebraic one, a second-order polynomial:

`2\cdot u^(2)-7\cdot u + 5 = 0`

Roots can be found by using the General Equation for Second-Order Polynomial:

`u = (7\pm √(49 - 40) )/(4)`

Roots are
`u_(1) = 2.5` and
`u_(2) = 1`. As tangent function has a periodicity of
`\pi`, solutions of
`u_(1)` and
`u_(2)` belong to first and third quadrants. Then, angles can be easily found by using inverse trigonometric functions:

`\theta_(1) = \tan^(-1) u_(1)`

`\theta_(1) \approx 0.379\cdot \pi` or
`\theta_(1) \approx 1.379\cdot \pi`

`\theta_(2) = \tan^(-1) u_(2)`

`\theta_(2) = 0.25\cdot \pi` or
`\theta_(2) = 1.25\cdot \pi`