Question

Find the possible values of $\sin x$, if $8\sin x - \cos x = 4$.

Answer 1

Hint: To solve this question, we will use some basic trigonometric identities. By using the identity, ${\sin ^2}x + {\cos ^2}x = 1$, we can simplify the given equation and then we can get the value of sin x.

Complete step-by-step answer:
Given,
$8\sin x - \cos x = 4$ ………………….……….. (i)
We have to find out all the possible values of sin x.
So,
Form equation (i),
$ \Rightarrow 8\sin x - \cos x = 4$
$ \Rightarrow 8\sin x - 4 = \cos x$ ……………. ………. (ii)
 We know that,
${\sin ^2}x + {\cos ^2}x = 1$.
Putting the value of cos x from equation (ii), we will get
$ \Rightarrow {\sin ^2}x + {\left( {8\sin x - 4} \right)^2} = 1$.
Solving this by using the identity, ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$, we will get
$
   \Rightarrow {\sin ^2}x + 64{\sin ^2}x + 16 - 64\sin x = 1 \\
   \Rightarrow 65{\sin ^2}x - 64\sin x + 15 = 0 \\
$
Solving the above quadratic equation by splitting the middle term, we will get
\[
   \Rightarrow 65{\sin ^2}x - \left( {39 + 25} \right)\sin x + 15 = 0 \\
   \Rightarrow 65{\sin ^2}x - 39\sin x - 25\sin x + 15 = 0 \\
   \Rightarrow 13\sin x\left( {5\sin x - 3} \right) - 5\left( {5\sin x - 3} \right) = 0 \\
   \Rightarrow \left( {13\sin x - 5} \right)\left( {5\sin x - 3} \right) = 0 \\
\]
Therefore,
$
  \left( {5\sin x - 3} \right) = 0 \\
\Rightarrow 5\sin x = 3 \\
$
$\sin x = \dfrac{3}{5}$
And,
\[
\Rightarrow \left( {13\sin x - 5} \right) = 0 \\
\Rightarrow 13\sin x = 5 \\
 \sin x = \dfrac{5}{{13}} \\
\]
Hence, the possible values of $\sin x$ in $8\sin x - \cos x = 4$ are $\sin x = \left( {\dfrac{3}{5},\dfrac{5}{{13}}} \right)$.

Note: Whenever we ask such types of questions, we have to use some basic trigonometric identities. First, we have to write the given equation in a simplified form and then according to that form, we will use the suitable identity which is useful. After that by using the identity, we are now able to make a quadratic equation in terms of trigonometric function. Now, we can solve that quadratic equation easily and by solving it, we will get the required possible answers.