Chapter 5.3, Problem 72E

To determine

To find : all the solutions of the equations in the given interval

Answer to Problem 72E

The solution to the quadratic equation $12{\sin }^{2}x-13\sin x+3=0$ in the interval $\left[0,2\pi \right)$ are $\underset{\_}{\boxed{x=0.3398,0.8481,2.2935,\&2.8018}}$ radians.

Explanation of Solution

Given information: $12{\sin }^{2}x-13\sin x+3=0$

Concept Involved:

Solution to a quadratic equation are the values of x that makes the equation TRUE. To solve a trigonometric equation, use standard algebraic techniques (when possible) such as collecting like terms, extracting square roots, and factoring. Our preliminary goal in solving a trigonometric equation is to isolate the trigonometric function on one side of the equation.

Formula Used:

If we have a quadratic equation of the form $a{x}^2+bx+c=0$ , then its solution is given by the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Let us substitute $u=\sin x$ in the equation $12{\sin }^{2}x-13\sin x+3=0$

  $12u^2-13u+3=0$

Identify the $a,b,\&c$ of the quadratic equation $a{u}^2+bu+c=0$

  $a=12;b=-13;c=3$

Substituting the values of a, b and c in the formula

  $u=\frac{-\left(-13\right)\pm \sqrt{{\left(-13\right)}^2-4\left(12\right)\left(3\right)}}{2\left(12\right)}$

Simplify the expression inside the square root

  $\begin{array}{l}u=\frac{13\pm \sqrt{169-144}}{24}\\ =\frac{13\pm \sqrt{25}}{24}\\ =\frac{13\pm 5}{24}\end{array}$

Find the two values of u $u=\frac{13+5}{24}$ & $u=\frac{13-5}{24}$

Combining the like terms in the numerator

  $u=\frac{18}{24}$ & $u=\frac{8}{24}$

Simplifying fraction to get the values of u $u=\frac{3}{4}$ & $u=\frac{1}{3}$

Back substitute $u=\sin x$

  $\sin x=\frac{3}{4}$ & $\sin x=\frac{1}{3}$

Label each equation

  $\sin x=\frac{3}{4}$▶1^st equation

  $\sin x=\frac{1}{3}$▶2^nd equation

Solving the 1^st equation $\sin x=3/4$ over the interval $\left[0,2\pi \right)$

  $x={\sin }^{-1}\left(3/4\right)$

  $x\approx 0.8481,2.2935$

Solving the 2^ndequation $\sin x=1/3$ over the interval $\left[0,2\pi \right)$

  $x={\sin }^{-1}\left(1/3\right)$

  $x\approx 0.3398,2.8018$