Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

Write the Number of Solutions of the Equation 4 Sin X − 3 Cos X = 7 - Mathematics

Sum

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

Solution

We have:
$4 \sin x - 3 \cos x = 7$
...(i)
The equation is of the form
$a \sin x + b \cos x = c$, where
$a = 4, b = - 3$ and $c = 7$
Now,
Let:
$a = r \sin \alpha$ and $a = r \sin \alpha$
Thus, we have:
$r = \sqrt{a^2 + b^2} = \sqrt{4^2 + 3^2} = 5$ and
$\tan \alpha = \frac{- 4}{3} \Rightarrow \alpha = \tan^{- 1} \left( - \frac{4}{3} \right)$
By putting $a = 4 = r \sin \alpha$ and $b = - 3 = r \cos \alpha$in equation (i), we get:
$r \sin\alpha \sin x + r \cos\alpha \cos x = 7$

$\Rightarrow r \cos (x - \alpha) = 7$

$\Rightarrow 5 \cos \left[ x - \tan^{- 1} \left( \frac{- 4}{3} \right) \right] = 7$

$\Rightarrow \cos \left[ x - \tan^{- 1} \left( \frac{- 4}{3} \right) \right] = \frac{7}{5}$

The solution is not possible.
Hence, the given equation has no solution.

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Q 2 | Page 26