Chapter 7.3, Problem 77AYU

To determine

To find: The trigonometric Quadratic equation in the form of ${\text{tan}}^2\theta =\left(3/2\right)\times \text{ sec}\theta ,0\le \theta <2\pi$.

Answer to Problem 77AYU

The solution set for the trigonometric Quadratic equation ${\text{tan}}^2\theta =\left(3/2\right)\times \text{ sec}\theta$.

$=\left\{\pi /3,5\pi /3\right\},0\le \theta <2\pi$

Explanation of Solution

Given:

Trigonometric Quadratic equation ${\text{tan}}^2\theta =\left(3/2\right)\times \text{ sec}\theta ,0\le \theta <2\pi$.

Calculation:

${\text{tan}}^2\theta =\left(3/2\right)\times \text{ sec}\theta$   -----1.

Using identities,

$1+{\text{ tan}}^2\theta ={\text{ sec}}^2\theta$

Therefore,

${\text{tan}}^2\theta ={\text{ sec}}^2\theta -1$

Using this formula in equation 1,

$={\text{ sec}}^2\theta -1=\left(3/2\right)\times \text{ sec}\theta$

Multiplying by 2 on each side,

$=2{\text{sec}}^2\theta -2=3\text{sec}\theta$

Subtracting $3\text{sec}\theta$ on each side,

$2{\text{sec}}^2\theta -3\text{sec}\theta -2=0$

Quadratic equation is in the form of $\text{sec}$ function can be factored.

$2{\text{sec}}^2\theta -4\text{sec}\theta +\text{ sec}\theta -2=0$

Taking the common factor $2\text{sec}\theta$ from $\left(2{\text{sec}}^2\theta -4\text{sec}\theta \right)$ and $+1$ from $\left(\text{sec}\theta -2\right)$,

$2\text{ sec}\theta \left(\text{sec}\theta -2\right)+1\left(\text{sec}\theta -2\right)=0$

Taking the common factor $\left(\text{sec}\theta -2\right)$,

$\left(\text{sec}\theta -2\right)\left(2\text{sec}\theta +1\right)$

Using zero product property,

$\left(\text{sec}\theta -2\right)=0$ or $\left(2\text{sec}\theta +1\right)$

$\text{sec}\theta =2$ or $\text{sec}\theta =-1/2$

Using the formula,

$\text{sec}\theta =1/\text{cos}\theta$

$1/\text{cos}\theta =2$ or $1/\text{cos}\theta =-1/2$

$\text{cos}\theta =\text{ 1}/2$ or $\text{cos}\theta =-2$

Angle for which $\cos$ function is $1/2=\pi /3,5\pi /3,0\le \theta <2\pi$.

Angle for which $\cos$ function is $-2=$ undefined, $0\le \theta <2\pi$.

The solution set $=\left\{\pi /3,5\pi /3\right\}$.