Question

Write the number of solutions of the trigonometric equation $4\sin x-3\cos x=7$.

Answer 1

Hint: The set of all the solutions of a given trigonometric equation constitute general solutions of the equation. You will try to find the solutions of such equations. These equations have one or more trigonometric ratios of unknown angles.

Complete step-by-step answer:

The equations that involve the trigonometric functions of a variable are called trigonometric equations. We will try to find the solutions of such equations. These equations have one or more trigonometric ratios of unknown angles. For example, $\cos x-\sin 2x=0$, is a trigonometric equation which does not satisfy all the values of x. Hence for such equations, we have to find the values of x or find the solution.


The given trigonometric equation is

$4\sin x-3\cos x=7$

Rearranging the terms, we get

$4\sin x-7=3\cos x$

Squaring on the both sides, we get

${{\left( 4\sin x-7 \right)}^{2}}={{\left( 3\cos x \right)}^{2}}$

Applying the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ on the left side, we get

$16{{\sin }^{2}}x-56\sin x+49=9{{\cos }^{2}}x$

We know that ${{\cos }^{2}}x=1-{{\sin }^{2}}x$ and replace in the right side, we get

$16{{\sin }^{2}}x-56\sin x+49=9\left( 1-{{\sin }^{2}}x \right)$

Rearranging the terms, we get

$16{{\sin }^{2}}x+9{{\sin }^{2}}x-56\sin x+49-9=0$

$25{{\sin }^{2}}x-56\sin x+40=0$

This is the quadratic equation in $\sin x$.

The discriminant =${{b}^{2}}-4ac$

 The discriminant = $3136-4000=-864<0$

So, we cannot find any real values for $\sin x$ and so for x.
Hence, there is no solution for the given trigonometric equation.

Note: Maximum value of any function of form $a\sin x-b\cos x=\sqrt{{{a}^{2}}+{{b}^{2}}}$. According to given question, Maximum value is$\sqrt{9+16}=\sqrt{25}=5$. Therefore, this is not possible. Hence there is no solution.