Question

The number of values of \[x\] in the interval\[[0,5\pi ]\]satisfying the equation \[3{\sin ^2}x - 7\sin x + 2 = 0\] is
A. \[0\]
B. \[5\]
C. \[6\]
D. \[10\]

Answer 1

Hints
In order to answer this question, we must first solve the trigonometric equation to obtain the values for x. To do this, we must first translate the equation into terms of the \[\tan x\] function using the trigonometric identity \[\frac{{\sin x}}{{\cos x}} = \tan x\]. Then, using the general solution for the \[\tan x\] function, which is given by \[n\pi + x\], where\[n \in Z\], we must substitute various values for n in order to obtain the necessary values in the interval.
Formula used:
\[\frac{{\sin x}}{{\cos x}} = \tan x\]
Complete step-by-step solution
We have been given the equation:
\[3{\sin ^2}x - 7\sin x + 2 = 0\]
Expand the above equation to factor:
\[ \Rightarrow 3{\sin ^2}x - 6\sin x - \sin x + 2 = 0\]
Factor the above equation:
\[ \Rightarrow (\sin x - 2)(3\sin x - 1) = 0\]
Solve for \[\sin x\]in\[(\sin x - 2) = 0\] by moving \[2\] to the other side of the equation:
Add \[2\] to both sides:
\[\sin \left( x \right) - 2 + 2 = 0 + 2\]
Simplify:
\[\sin \left( x \right) = 2\]
Solve for \[\sin x\] in\[(3\sin x - 1) = 0\]:
Simplify:
\[{\rm{ }}3\sin \left( x \right) = 1\]
Divide either side of the equation by \[3\]:
\[\frac{{3\sin \left( x \right)}}{3} = \frac{1}{3}\]
Simplify:
\[\sin \left( x \right) = \frac{1}{3}\]
But, \[\sin x\] cannot be more than \[1\].
\[\therefore \sin x = 1/3\]
\[\sin x\] =2 is not possible
It means, value of \[\sin x\] is positive.
It means,\[x\] lies in first and second quadrant.
So, there are \[6\] values from \[0\] to \[2\pi \].
Then \[\alpha ,|\pi - \alpha |,2\pi + \alpha ,|3\pi - \alpha |,4\pi + \alpha \mid 5\pi - \alpha \] are the solutions in \[[0,5\pi ]\]
Hence, the option C is correct.
Note
A trigonometric equation is one that uses one or more trigonometric ratios of an unknown angle. A trigonometric identity isn't the same as a trigonometric equation. Every value of the unknown angle results in the satisfaction of an identity, and for some specific values of the unknown angle, the satisfaction of a trigonometric equation. The unknown angle's solution is the value of the angle that answers the trigonometric equation.