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Trigonometrical Equations
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$tan\, (5\pi\, cos\, \theta ) = cot (5 \pi \,sin\, \theta )$ માટે $\theta$ ની $(0, 2\pi )$ માં ઉકેલોની સંખ્યા ........... થાય
A
$28$
B
$14$
C
$4$
D
$2$
Solution
$tan \,(5\pi \,cos\, \theta ) = cot\, (5 \pi\, sin\, \theta )$
$tan \,(5\pi \,cos\, \theta ) = tan\, \left( {\frac{\pi }{2}\, – \,5\pi \sin \theta } \right)$
$5\pi \,cos\theta = n\pi + \pi /2 – 5\pi \,sin\theta$
$(cos\theta + sin\theta ) = \left( {\frac{{2n + 1}}{{10}}} \right)$
$\Rightarrow -1 < \frac{{2n + 1}}{{10\sqrt 2 }} < 1$
$\Rightarrow$ $ – \,\frac{{10\sqrt 2 \, – 1}}{2}$ $< n <$ $\frac{{10\sqrt 2 \, – 1}}{2}$
$\therefore$ $n= 14$ for each $‘n’$ there are two values of $\theta$
no. of solutions $= 28$
Standard 11
Mathematics
