Question

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

Answer 1

Hint: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.