Mathematics: Find all exact solutions on [0, 2π). 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 :)

Find all exact solutions on [0, 2π). I tried using

batmanmarie2004

21.01.2022

$\displaystyle t=\frac{\pi}{3},\ t=\frac{5\pi}{3}$

Step-by-step 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

$\displaystyle sect=2,\ sect=-\frac{1}{2}$

Recalling

$\displaystyle cost=\frac{1}{sect}$

The solutions are also expressed as

$\displaystyle cost=\frac{1}{2},\ cost=-2$

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]

$\boxed{\displaystyle t=\frac{\pi}{3},\ t=\frac{5\pi}{3}}$

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