Download App

INVERSE TRIGNOMETRIC FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINVERSE TRIGNOMETRIC FUNCTIONS3 Marks
Question
Write the following in the simplest form:
$\tan^{-1}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}},-\text{a}<\text{x}<\text{a}$
✓

Answer

Let, $\text{x}=\text{a}\cos\theta$
Now,
$\tan^{-1}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}=\tan^{-1}\sqrt{\frac{\text{a}-\text{a}\cos\theta}{\text{a}+\text{a}\cos\theta}}$
$=\tan^{-1}\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$
$=\tan^{-1}\sqrt{\frac{2\sin^2\frac{\theta}{2}}{2\cos^2\frac{\theta}{2}}}$
$=\tan^{-1}\Big(\tan\frac{\theta}{2}\Big)$
$=\frac{\theta}{2}$
$=\frac{1}{2}\cos^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)$
$\therefore\ \tan^{-1}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}=\frac{\cos^{-1}\big(\frac{\text{x}}{\text{a}}\big)}{2}$

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 Question" 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

Find which of the binary operations are commutative and which are associative.
Let A = N × N and * be the binary operation on A defined by:
(a, b) * (c, d) = (a + c, b + d)
→
Find the general solution of $y d x+\left(x-y^{2}\right) d y=0$
→
Given $\text{A}=\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix},$ compute $A^{-1}$ and show that $2A^{-1} = 9I - A$.
→
If $g(x)=x^2+x-2$ and $(g o f)(x)=4 x^2-10 x +4 ,$ find $f(x)$.
→
Solve : $\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}$.
→
Classify the following functions as injection, surjection or bijection:f: $R \rightarrow R$, defined by $f(x)=1+x^2$
→
Write a value of $\int\frac{(\log\text{x})^{\text{n}}}{\text{x}}\text{ dx}$
→
Solve the following equation for x:
$\tan^{-1}(2+\text{x})+\tan^{-1}(2-\text{x})=\tan^{-1}\frac{2}{3},$ where $\text{x}<-\sqrt3$ or, $\text{x}>\sqrt3$
→
Write the coordinates of the point on the curve $\text{x}=\text{e}^\text{t}\cos\text{t},\text{y}=\text{e}^\text{t}$ where the tangent line makes an angle $\frac{\pi}{4}$ with x-axis.
→
For each of the differential equations given in find the general solution:
$(\text{x}+3\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}(\text{y}>0)$
→