Answer:
[tex]\displaystyle t=\frac{\pi}{3},\ t=\frac{5\pi}{3}[/tex]
Step-by-step explanation:
Trigonometric Equations
The given equation is
[tex]2tan^2t=3sect[/tex]
To solve the equation we must recall that
[tex]tan^2t=sec^2t-1[/tex]
Substituting into the given equation
[tex]2(sec^2t-1)=3sect[/tex]
Operating and rearranging
[tex]2sec^2t-3sect-2=0[/tex]
This is a second-degree equation in terms of sect. Factoring
[tex](sect-2)(2sect+1)=0[/tex]
This produces two solutions
[tex]\displaystyle sect=2,\ sect=-\frac{1}{2}[/tex]
Recalling
[tex]\displaystyle cost=\frac{1}{sect}[/tex]
The solutions are also expressed as
[tex]\displaystyle cost=\frac{1}{2},\ cost=-2[/tex]
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]
[tex]\boxed{\displaystyle t=\frac{\pi}{3},\ t=\frac{5\pi}{3}}[/tex]
