Question

Find all exact solutions on [0, 2π).


`2tan^2(t) = 3 sec(t)`

I tried using some trig identities only to get lost or end up getting some complicated equation! Any help is very much appreciated, thank you :)

Answer 1

`\displaystyle t=(\pi)/(3),\ t=(5\pi)/(3)`

Explanation:

Trigonometric Equations

The given equation is

`2tan^2t=3sect`

To solve the equation we must recall that

`tan^2t=sec^2t-1`

Substituting into the given equation

`2(sec^2t-1)=3sect`

Operating and rearranging

`2sec^2t-3sect-2=0`

This is a second-degree equation in terms of sect. Factoring

`(sect-2)(2sect+1)=0`

This produces two solutions

`\displaystyle sect=2,\ sect=-(1)/(2)`

Recalling

`\displaystyle cost=(1)/(sect)`

The solutions are also expressed as

`\displaystyle cost=(1)/(2),\ cost=-2`

since the magnitude of the cosine cannot be greater than 1, the only acceptable solution is

\displaystyle cost=\frac{1}{2}

Which has two possible angles in the interval [0,2\pi]

`\boxed{\displaystyle t=(\pi)/(3),\ t=(5\pi)/(3)}`