Question

Find all exact solutions on [0, 2π).

Answer 1

Answer:

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

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]

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