Question

The equation 3 cos x +4 sin x=6 has .... solution.

• A. finite
• B. infinite
• C. one
• D. no

Answer 1

The correct option is D

no

$Givenequation:3cosx+4sinx=6..\left(i\right)\text{Thus, the equation is of the form a cos x=b~sin~x=c,where~a~=3,b=4~and~c=6.}Let:a=3=rcos\alpha andb-4=rsin\alpha Now,tab\alpha =\frac{b}{a}=\frac{4}{5}\Rightarrow \alpha =ta{n}^{-1}\left(\frac{4}{3}\right)Also,r=\sqrt{a^2+b^2}=\sqrt{9+16}=\sqrt{2}5=5Onputtinga=3=rcos\alpha andb=4=rsin\alpha andb=4=rsin\alpha inequation\left(i\right),weget:rcos\alpha cos\theta +sin\alpha sin\theta =6\Rightarrow rcos(\theta -\alpha )=6\Rightarrow 5cos(\theta -\alpha )=6\Rightarrow cos(\theta -\alpha )=\frac{6}{5}\text{From here,we cannot find the value of}\theta .$