For the principal values, evaluate the following: ^ -1 [ 2cosec^ … - Vidyadip

Question

For the principal values, evaluate the following:
$\sin^{-1}[\cos\{2\text{cosec}^{-1}(-2)\}]$

✓

Answer

$\text{cosec}^{-1}\text{x}$ represents an angle in $\Big[-\frac{\pi}{2},0\Big)\cup\Big(0,\frac{\pi}{2}\Big]$ whose cosecant is x.

Let $\text{x}=\text{cosec}^{-1}(-2)$

$\Rightarrow\text{cosec x}=-2=\text{cosec c}\Big(-\frac{\pi}{6}\Big)$

$\Rightarrow\text{x}=-\frac{\pi}{6}$

$\sin^{-1}[\cos\{2\text{cosec}^{-1}(-2)\}]=\sin^{-1}\Big[\cos\Big\{2\times\Big(-\frac{\pi}{6}\Big)\Big\}\Big]$

$=\sin^{-1}\Big[\cos\Big(-\frac{\pi}{3}\Big)\Big]=\sin^{-1}\Big[\frac{1}{2}\Big]$

$\sin^{-1}\text{x}$ represents an angle in $\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]$ whose sin is x.

Let $\text{x}=\sin^{-1}\Big[\frac{1}{2}\Big]$

$\Rightarrow\sin\text{x}=\frac{1}{2}=\sin\Big(\frac{\pi}{6}\Big)$

$\Rightarrow\text{x}=\frac{\pi}{6}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from INVERSE TRIGNOMETRIC FUNCTIONS→INVERSE TRIGNOMETRIC FUNCTIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\log\tan\text{x dx}$

Two dice are rolled once. Find the probability that:

the numbers on two dice are different.
the total of numbers on the two dice is at least.

Compute $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big),$ if P(B) = 0.5 and $\text{P}(\text{A}\cap\text{B})=0.32$

There are 3 red and 5 black balls in bag 'A'; and 2 red and 3 black balls in bag 'B'. One ball is drawn from bag 'A' and two from bag 'B'. Find the probability that out of the 3 balls drawn one is red and 2 are black.

Find the values of p so that the lines $\frac{1-\text{x}}{3}=\frac{7\text{y}-14}{2\text{p}}=\frac{\text{z}-3}{2}\ \text{and}\ \frac{7-7\text{x}}{3\text{p}}=\frac{\text{y}-5}{1}=\frac{6-\text{z}}{5}$ are at right angles.

Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

Evaluate the following integrals:

$\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{2})\text{dx}$

If $\text{A}=\text{diag}\begin{pmatrix}2&-5&9\end{pmatrix},\text{ B}=\text{diag}\begin{pmatrix}1&1&-4\end{pmatrix}$ and $\text{C}=\text{diag}\begin{pmatrix}-6&3&4\end{pmatrix},$ find.
$\text{B}+\text{C}-2\text{A}$

Show that: $\tan\bigg(\frac{1}{2}\sin^{-1}\frac{3}{4}\bigg) = \frac{4-\sqrt{7}}{3}.$

Find the slope of the normal at the point 't' on the curve $\text{x}=\frac{1}{\text{t}},\text{y}=\text{t}.$