Question

The equation $3\cos x+4\sin x=6$ has .... solution.
(a) finite
(b) infinite
(c) one
(d) no

Answer 1

(d) no
Given equation:
$3\cos x+4\sin x=6$ ...(i)
Thus, the equation is of the form $a\cos x+b\sin x=c$, where $a=3,b=4$ and $c=6$.
Let:
$a=3=r\cos \alpha$ and $b=4=r\sin \alpha$
Now,
$$\begin{gathered} \tan \alpha =\frac{b}{a}=\frac{4}{3} \\ \Rightarrow \alpha ={\tan }^{-1}\left(\frac{4}{3}\right) \end{gathered}$$
Aso,
$r=\sqrt{a^2+b^2}=\sqrt{9+16}=\sqrt{25}=5$

On putting $a=3=r\cos \alpha$ and$b=4=r\sin \alpha$ in equation (i), we get:
$$\begin{gathered} r\cos \alpha \cos \theta +\sin \alpha \sin \theta =6 \\ \Rightarrow r\cos (\theta -\alpha )=6 \\ \Rightarrow 5\cos (\theta -\alpha )=6 \\ \Rightarrow \cos (\theta -\alpha )=\frac{6}{5} \end{gathered}$$
From here, we cannot find the value of $\theta$.