recategorized by
362 views
0 votes
0 votes

If $\sin^{-1}\frac{1}{\sqrt{5}}$ and $\cos^{-1}\frac{3}{\sqrt{10}}$ lie in $\left[0,\frac{\pi}{2}\right],$ their sum is equal to

  1. $\frac{\pi}{6}$
  2. $\frac{\pi}{3}$
  3. $\sin^{-1}\frac{1}{\sqrt{50}}$
  4. $\frac{\pi}{4}$
recategorized by

1 Answer

1 votes
1 votes

Answer: $\textbf D$

Let: $\bf {\alpha} = \sin ^{-1}\frac{1}{\sqrt 5}$, and $\bf {\beta} = \cos^{-1} \frac{3}{\sqrt {10}}$


$\therefore\mathbf{ \sin \alpha} = \frac{1}{\sqrt 5} = \frac{\bf P}{\bf H}, \; \cos \beta = \frac{3}{\sqrt{10}}= \frac{\bf B} {\bf H}$

By using Pythagoras Theorem:

$(\because \mathrm {P^2 + B^2 = H^2 \Rightarrow B = \sqrt {H^2 - P^2}} = \sqrt {5-1} = 2)$

$\Rightarrow \cos \alpha = \frac{2}{\sqrt 5}$

Similarly,

$\Rightarrow \cos \beta = \frac{3}{\sqrt {10}}$

$\Rightarrow \sin \beta = \frac{1}{\sqrt {10}}$

Now, We know that:

$\mathrm {\sin (A+B) = \sin A \cos B + \cos A \sin B}$

$\begin {align}\Rightarrow \sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta\\ &= \frac{1}{\sqrt 5}.\frac{3}{\sqrt {10}} + \frac{2}{\sqrt {5}}. \frac{1}{\sqrt{10}} \\ &= \frac{3}{\sqrt{50}} + \frac{2}{\sqrt{50}}\\&=\frac{5}{\sqrt{50}}\\&= \frac{5}{5\sqrt 2} \\&=\frac{1}{\sqrt 2}\end {align}$

$\begin {align} \Rightarrow \alpha + \beta &= \sin ^{-1} \frac{1}{\sqrt 2} \\&= \sin ^{-1} \Bigg({\sin \frac{\pi}{4}}\Bigg) \\&= \frac{\pi}{4}\end {align}$

Answer $\therefore \textbf D $ is the correct option.

edited by

Related questions

0 votes
0 votes
2 answers
1
gatecse asked Sep 18, 2019
435 views
If $\tan\: x=p+1$ and $\tan\; y=p-1,$ then the value of $2\:\cot\:(x-y)$ is$2p$$p^{2}$$(p+1)(p-1)$$\frac{2p}{p^{2}-1}$
0 votes
0 votes
0 answers
2
gatecse asked Sep 18, 2019
272 views
The equations $x=a\cos\theta+b\sin\theta$ and $y=a\sin\theta+b\cos\theta,(0\leq\theta\leq2\pi$ and $a,b$ are arbitrary constants$)$ representa circlea parabolaan ellipsea...
0 votes
0 votes
0 answers
3
gatecse asked Sep 18, 2019
340 views
If in a $\triangle ABC,\angle B=\dfrac{2\pi}{3},$ then $\cos A+\cos C$ lies in$\left[\:-\sqrt{3},\sqrt{3}\:\right]$$\left(\:-\sqrt{3},\sqrt{3}\:\right]$$\left(\:\frac{3}{...
0 votes
0 votes
1 answer
4
gatecse asked Sep 18, 2019
241 views
The value of $\sin^{6}\frac{\pi}{81}+\cos^{6}\frac{\pi}{81}-1+3\sin^{2}\frac{\pi}{81}\:\cos^{2}\frac{\pi}{81}$ is$\tan^{6}\frac{\pi}{81}$$0$$-1$None of these