Question

How do you solve $6{{\sin }^{2}}x+\sin x-1=0$?

Answer 1

In the given question, we have been asked to solve the trigonometric equation. In order to solve the equation, we substitute $\sin x$ by $u$, and ${{\sin }^{2}}x$ by ${{u}^{2}}$in the given equation as $6{{u}^{2}}+u-1=0$. From here, we take out common factors and equate each factor with zero and solve further to get the required solution.

Complete step by step answer:
We have the given equation: $6{{\sin }^{2}}x+\sin x-1=0$
Replace $\sin x$ by $u$, and ${{\sin }^{2}}x$ by ${{u}^{2}}$ in the given equation, we obtain
$6{{u}^{2}}+u-1=0$
It will give us a quadratic equation.Splitting the middle term of the above quadratic equation, we get, $6{{u}^{2}}+3u-2u-1=0$.
Take a common factor from each bracket after forming a pair,, we get
$3u\left( 2u+1 \right)-1\left( 2u+1 \right)=0$
Take out the common factor, we get
$\left( 3u-1 \right)\left( 2u+1 \right)=0$
Equate each factors equals to $0$, we get
$3u-1=0$ and $2u+1=0$
By simplifying both the common factor, we get
$u=\dfrac{1}{3}$ and $u=-\dfrac{1}{2}$
Now, replace $u$ by $\sin x$
Therefore, we get
$\sin x=\dfrac{1}{3}$ and $\sin x=-\dfrac{1}{2}$
When, $\sin x=\dfrac{1}{3}$
Taking ${{\sin }^{-1}}$ on both the side of the above equation, we get
${{\sin }^{-1}}\left( \sin x \right)={{\sin }^{-1}}\dfrac{1}{3}$
$x={{\sin }^{-1}}\dfrac{1}{3}$
With the help of using calculator,
$x=0.3398$
And the sine function is positive in the first and the second quadrant.
To find second solution, subtract reference angle from $\pi$ i.e. 3.1415
$x=3.1415-0.3398=2.8017$
Therefore, the solution of $x$ are,
$\Rightarrow x=0.3398\,and\ 2.8017$
When, $\sin x=-\dfrac{1}{2}$
Here $\sin x$ is negative in the third and the fourth quadrants. Since,$\dfrac{\sin \pi }{6}=\dfrac{1}{2}$
$\sin x$ would be $-\dfrac{1}{2}$ for
$x=\pi +\dfrac{\pi }{6}$ and $x=2\pi -\dfrac{\pi }{6}$
Therefore, the solution of $x$ are,
$x=\dfrac{7\pi }{6}\ and\ \dfrac{11\pi }{6}$
Converting into decimal with the help of calculator by putting the value of $\pi =3.1415$, we get
$\Rightarrow x=3.6651\ and\ 5.7595$

The values of $x$ are $0.3398$, $2.8017$, $3.6651$, and $5.7595$.

Note: In the given question, we need to find the correct formula to solve the given equation. After finding the formula, we should expand that and it will give us a quadratic equation and after splitting the middle term, find the solution of $x$. In order to solve these types of questions, you need to know about the basic trigonometric subject and how to solve quadratic equations.