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Trigonometrical Equations
medium
If $\sin \theta + \cos \theta = 1$ then the general value of $\theta $ is
A
$2n\pi $
B
$n\pi + {( - 1)^n}\frac{\pi }{4} - \frac{\pi }{4}$
C
$2n\pi + \frac{\pi }{2}$
D
None of these
(IIT-1981)
Solution
(b) $\sin \theta + \cos \theta = 1$
$\Rightarrow \frac{1}{{\sqrt 2 }}\sin \theta + \frac{1}{{\sqrt 2 }}\cos \theta = \frac{1}{{\sqrt 2 }}$
Dividing by $\sqrt {{1^2} + {1^2}} = \sqrt 2 $, we get
$\sin \left( {\theta + \frac{\pi }{4}} \right) = \frac{1}{{\sqrt 2 }} = \sin \frac{\pi }{4}$
$ \Rightarrow $ $\theta + \frac{\pi }{4} = n\pi + {( – 1)^n}\frac{\pi }{4}$
$\Rightarrow \theta = n\pi + {( – 1)^n}\frac{\pi }{4} – \frac{\pi }{4}$.
Standard 11
Mathematics
