Question

$\mathrm{If}\mathrm{cosp\theta }+\mathrm{cosq\theta }=0,\mathrm{then}\mathrm{the}\mathrm{different}\mathrm{values}\mathrm{of}\theta \mathrm{are}\mathrm{in}A.P.\mathrm{with}a\mathrm{common}\mathrm{difference}$
.

• A. $2\pi /\left(p\pm q\right)$
• B. $\pi /\left(p\pm q\right)$
• C. $3\pi /\left(p\pm q\right)$

Answer 1

The correct option is A $2\pi /\left(p\pm q\right)$

Here, we can solve the equation as:
$\cos p\theta +\cos q\theta =0\Rightarrow \cos p\theta =-\cos q\theta \Rightarrow \cos p\theta =\cos (\pi -q\theta )\mathrm{Thus},\mathrm{the}\mathrm{gen}e\mathrm{ral}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{can}\mathrm{be}\mathrm{given}\mathrm{as}:p\theta =2n\pi \pm (\pi -\mathrm{q\theta }),n\in Z.\Rightarrow (p\mp q)\theta =(2n+1)\pi ,n\in Z.\Rightarrow \theta =\frac{(2n+1)\pi }{(p\mp q)},n\in Z.\mathrm{Now},\mathrm{putting}\mathrm{values}\mathrm{of}n,\mathrm{we}\mathrm{get}\mathrm{values}\mathrm{of}\theta =\frac{\pi }{(p\mp q)},\frac{3\pi }{(p\mp q)},..\mathrm{clearly}\mathrm{these}\mathrm{solutions}\mathrm{form}\mathrm{an}A.P.\mathrm{with}\mathrm{common}\mathrm{difference}=\frac{2\pi }{(p\pm q)}.$