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Trigonometrical Equations
medium
If $\sqrt 2 \sec \theta + \tan \theta = 1,$ then the general value $\theta $ is
A
$n\pi + \frac{{3\pi }}{4}$
B
$2n\pi + \frac{\pi }{4}$
C
$2n\pi - \frac{\pi }{4}$
D
$2n\pi \pm \frac{\pi }{4}$
Solution
(c) $\sqrt 2 \sec \theta + \tan \theta = 1 $
$\Rightarrow \frac{{\sqrt 2 }}{{\cos \theta }} + \frac{{\sin \theta }}{{\cos \theta }} = 1$
$ \Rightarrow $ $\sin \theta – \cos \theta = – \sqrt 2 $
Dividing by $\sqrt 2 $ on both sides, we get
$\frac{1}{{\sqrt 2 }}\sin \theta – \frac{1}{{\sqrt 2 }}\cos \theta = – 1$
$ \Rightarrow $ $\frac{1}{{\sqrt 2 }}\cos \theta – \frac{1}{{\sqrt 2 }}\sin \theta = 1 $
$\Rightarrow \cos \,\left( {\theta + \frac{\pi }{4}} \right) = \cos (0)$
$ \Rightarrow $ $\theta + \frac{\pi }{4} = 2n\pi \pm 0$
$\Rightarrow \theta = 2n\pi – \frac{\pi }{4}$.
Standard 11
Mathematics
