ncert-class12-day9-chap2-inverse-trigonometric-functions-theory-part-1-

<p style="font-size:40px ; color:red">
<marquee behavior="scroll" direction="left">Acknowledgement and Disclaimer</marquee></p>

$\quad$ $\quad$ **Acknowledgements and Disclaimer !!**

• The images/contents are used for teaching purpose. The copyright remains with the original creator. If you suspect a violation, bring it to my notice and we will remove that part...

</div>

$\quad$ $\qquad$ **अभिस्वीकृति और अस्वीकरण !!**

- चित्रों/सामग्री का उपयोग शिक्षण उद्देश्य के लिए किया जाता है। कॉपीराइट मूल निर्माता के पास रहता है। यदि आपको किसी उल्लंघन का संदेह है, तो इसे हमारे ध्यान में लाएं और हम उस हिस्से को हटा देंगे...

</div>

======

$\quad$

### <Primary>Inverse Trigonometric Functions</Primary>

======
<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Outline</B></Primary>

- Basic Concepts

- $\sin^{-1} $ Function

- Graph of $\sin^{-1} $ Function

- $\cos^{-1} $ Function

- Graph of $\cos^{-1} $ Function

- $\cosec^{–1}$ Function

- Graph of  $\cosec^{-1}$ Function

</div>

- $\sec ^{-1}$ Function

- $\tan ^{-1}$ Function

- Graph of $\tan ^{-1}$ Function

- $\cot ^{-1}$ Function

- Graph of $\cot ^{-1}$ Function

- Note

- Examples

</div>

- बुनियादी अवधारणाएं

- $\sin^{-1} $ फलन

- $\sin^{-1} $ फलन का ग्राफ

- $\cos^{-1} $ फलन

- $\cos^{-1} $ फलन का ग्राफ

- $\cosec^{-1}$ (कोसेकेंट इनवर्स) फलन

- $\cosec^{-1}$ फलन का ग्राफ

</div>

- $\sec^{-1}$ (सेकेंट इनवर्स) फलन

- $\tan^{-1}$ (टैंगेंट इनवर्स) फलन

- $\tan^{-1}$ फलन का ग्राफ

- $\cot^{-1}$ (कोटंगेंट इनवर्स) फलन

- $\cot^{-1}$ फलन का ग्राफ

- नोट

- उदाहरण

</div>

======
<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Basic Concepts</B></Primary>

<hr>

<main>

<table>
        <caption>Domain and Range of Trigonometric Functions</caption>
        <thead>
            <tr>
                <th>Function</th>
                <th>Domain (in radians)</th>
                <th>Range</th>
            </tr>
        </thead>
        <tbody>
            <tr>
                <td>Sine (sin(x))</td>
                <td>All real numbers (-∞, ∞)</td>
                <td>[-1, 1]</td>
            </tr>
            <tr>
                <td>Cosine (cos(x))</td>
                <td>All real numbers (-∞, ∞)</td>
                <td>[-1, 1]</td>
            </tr>
            <tr>
                <td>Tangent (tan(x))</td>
                <td>x ≠ (π/2) + kπ, where k is any integer</td>
                <td>All real numbers (-∞, ∞)</td>
            </tr>
            <tr>
                <td>Cotangent (cot(x))</td>
                <td>x ≠ kπ, where k is any integer</td>
                <td>All real numbers (-∞, ∞)</td>
            </tr>
            <tr>
                <td>Secant (sec(x))</td>
                <td>x ≠ (π/2) + kπ, where k is any integer</td>
                <td>All real numbers except for (-∞, -1] and [1, ∞)</td>
            </tr>
            <tr>
                <td>Cosecant (csc(x))</td>
                <td>x ≠ kπ, where k is any integer</td>
                <td>All real numbers except for (-∞, -1] and [1, ∞)</td>
            </tr>
        </tbody>
    </table>

</div>

</main>

<hr>

<footer> <span style="color:blue">Basic Concepts</span> $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Basic Concepts</B></Primary>

<hr>

<main>

sine function, i.e.,

sine : $\mathbf{R} \rightarrow[-1,1]$

cosine function, i.e.,

$\cos : \mathbf{R} \rightarrow[-1,1]$

tangent function, i.e.,

$\tan : \mathbf{R}-\\{x: x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

sine फलन, अर्थात्,

$\sin : \mathbf{R} \rightarrow[-1,1]$

cosine फलन, अर्थात्,

$\cos : \mathbf{R} \rightarrow[-1,1]$

tangent फलन, अर्थात्,

$\tan : \mathbf{R}-\\{x: x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}$

</div>

</main>

<hr>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Basic Concepts</B></Primary>

<hr>

<main>

cotangent function, i.e.,

$\cot : \mathbf{R}-\\{x: x=n \pi, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}$

secant function, i.e.,

sec : $\mathbf{R}-\\{x: x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}-(-1,1)$

cosecant function, i.e.,

$cosec: \mathbf{R}-\\{x: x=n \pi, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}-(-1,1)$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

cotangent फलन, अर्थात्

$\cot : \mathbf{R}-\\{x: x=n \pi, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}$

secant फलन, अर्थात्

$\mathbf{R}-\\{x: x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}-(-1,1)$

cosecant फलन, अर्थात्

$cosec: \mathbf{R}-\\{x: x=n \pi, n \in \mathbf{Z}\\} \rightarrow \mathbf{R}-(-1,1)$

</div>

</main>

<hr>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\sin^{-1} $ Function</B></Primary>

<hr>

<main>

The branch with range ,$[-π/2, π/2]$ is called the principal value branch,
whereas other intervals as range give different branches of $\sin^{-1} $.

When we refer
to the function $\sin^{-1} $, we take it as the function whose domain is [–1, 1] and range is $[-π/2, π/2]$.

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

वह शाखा, जिसका परिसर $[-π/2, π/2]$ है, मुख्य शाखा (मुख्य मान शाखा) कहलाती है, जब कि परिसर के रूप में अन्य अंतरालों से sin' की भिन्न-भिन्न शाखाएँ मिलती हैं।

जब हम फलन sin का उल्लेख करते हैं, तब हम इसे प्रांत [-1, 1] तथा परिसर $[-π/2, π/2]$

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  <span style="color:blue">$\sin^{-1} $ Function</span> $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\sin^{-1} $ Function</B></Primary>

<hr>

<main>

From the definition of the inverse functions, it follows that sin ($\sin^{-1} $ x) = x
if – 1 ≤ x ≤ 1 and $\sin^{-1} $ (sin x) = x if x in $[-π/2, π/2]$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

प्रतिलोम फलन की परिभाषा द्वारा, यह निष्कर्ष निकलता है कि sin (sin x) = x, यदि 1≤x4 तथा sin¯¹ (sin x) = x यदि $[-π/2, π/2]$

</div>

</main>

<hr>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Graph of $\sin^{-1} $ Function</B></Primary>

<hr>

<main>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ <span style="color:blue">Graph of $\sin^{-1} $ Function</span> $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\cos^{-1} $ Function</B></Primary>

<hr>

<main>

The branch with range [0, π] is called the principal
value branch of the function $\cos^{-1} $

We write $\cos^{-1} $ : [–1, 1] → [0, π].

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

वह शाखा, जिसका परिसर [0, ] है, मुख्य शाखा (मुख्य मान शाखा) कहलाती है और हम लिखते हैं कि

cos¹: [-1, 1] →[0, π]

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  <span style="color:blue">$\cos^{-1} $ Function</span>  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Graph of $\cos^{-1} $ Function</B></Primary>

<hr>

<main>

</div>
<div  style='float: right; width:44%; text-align: left;font-size:21px'>

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ <span style="color:blue">Graph of $\cos^{-1} $ Function</span>   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\cosec^{–1}$ Function</B></Primary>

<hr>

<main>

$\cosec^{-1}$ can be defined as a function whose domain is $\mathbf{R}-(-1,1)$ and range could be any of the intervals $\frac{-3 \pi}{2}, \frac{-\pi}{2}-\\{-\pi\\}, \frac{-\pi}{2}, \frac{\pi}{2}-\\{0\\},[\frac{\pi}{2}, \frac{3 \pi}{2}]-\\{\pi\\}$ etc. The function corresponding to the range $[\frac{-\pi}{2}, \frac{\pi}{2}]-\\{0\\}$ is called the principal value branch of $cosec^{-1}$. We thus have principal branch as

$$
cosec^{-1}: \mathbf{R}-(-1,1) \rightarrow[\frac{-\pi}{2}, \frac{\pi}{2}]-\\{0\\}
$$
</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

cosec¹ एक ऐसे फलन के रूप में परिभाषित हो सकता है जिसका प्रांत R- (-1, 1) है और परिसर
-3π -π
अंतराल  $\frac{-3 \pi}{2}, \frac{-\pi}{2}-\\{-\pi\\}, \frac{-\pi}{2}, \frac{\pi}{2}-\\{0\\},[\frac{\pi}{2}, \frac{3 \pi}{2}]-\\{\pi\\}$
इत्यादि में से कोई भी एक हो सकता है। परिसर $[\frac{-\pi}{2}, \frac{\pi}{2}]-\\{0\\}$ के संगत फलन मुख्य शाखा कहते हैं।

मुख्य शाखा निम्नलिखित तरह से व्यक्त होती है

$$
cosec^{-1}: \mathbf{R}-(-1,1) \rightarrow[\frac{-\pi}{2}, \frac{\pi}{2}]-\\{0\\}
$$
</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   <span style="color:blue">$\cosec^{–1}$ Function</span> $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Graph of  $\cosec^{-1}$ Function</B></Primary>

<hr>

<main>

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  <span style="color:blue">Graph of  $\cosec^{-1}$ Function</span>   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\sec ^{-1}$ Function</B></Primary>

<hr>

<main>

$\sec ^{-1}$ can be defined as a function whose domain is $\mathbf{R}-(-1,1)$ and range could be any of the intervals $[-\pi, 0]-\\{\frac{-\pi}{2}\\},[0, \pi]-\\{\frac{\pi}{2}\\},[\pi, 2 \pi]-\\{\frac{3 \pi}{2}\\}$ etc. Corresponding to each of these intervals, we get different branches of the function $sec^{-1}$. The branch with range $[0, \pi]-\\{\frac{\pi}{2}\\}$ is called the principal value branch of the function $sec^{-1}$. We thus have

$$
\sec ^{-1}: \mathbf{R}-(-1,1) \rightarrow[0, \pi]-\\{\frac{\pi}{2}\\}
$$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

sec¹ एक ऐसे फलन के रूप में परिभाषित हो सकता है जिसका प्रांत (-1, 1) हो और जिसका परिसर अंतरालों   $[-\pi, 0]-\\{\frac{-\pi}{2}\\},[0, \pi]-\\{\frac{\pi}{2}\\},[\pi, 2 \pi]-\\{\frac{3 \pi}{2}\\}$ इनमें से प्रत्येक अंतराल के संगत हमें फलन sec-¹ की भिन्न-भिन्न शाखाएँ प्राप्त  हैं। वह शाखा जिसका परिसर   $[0, \pi]-\\{\frac{\pi}{2}\\}$ होता है,  फलन sec की  इसको हम निम्नलिखित प्रकार से व्यक्त करते हैं:

$$
\sec ^{-1}: \mathbf{R}-(-1,1) \rightarrow[0, \pi]-\\{\frac{\pi}{2}\\}
$$

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  <span style="color:blue">$\sec ^{-1}$ Function</span> $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Graph of $\sec ^{-1}$ Function</B></Primary>

<hr>

<main>

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  <span style="color:blue">$\sec ^{-1}$ Function</span> $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\tan ^{-1}$ Function</B></Primary>

<hr>

<main>

$\tan ^{-1}$ can be defined as a function whose domain is $\mathbf{R}$ and range could be any of the intervals $(\frac{-3 \pi}{2}, \frac{-\pi}{2}),(\frac{-\pi}{2}, \frac{\pi}{2}),(\frac{\pi}{2}, \frac{3 \pi}{2})$ and so on. These intervals give different branches of the function $\tan ^{-1}$. The branch with range $(\frac{-\pi}{2}, \frac{\pi}{2})$ is called the principal value branch of the function $\tan ^{-1}$.

We thus have

$$
\tan ^{-1}: \mathbf{R} \rightarrow(\frac{-\pi}{2}, \frac{\pi}{2})
$$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

tan¹ एक
ऐसे फलन के रूप में  परिभाषित  सकता है, जिसका प्रांत R हो और परिसर अंतरालों  $(\frac{-3 \pi}{2}, \frac{-\pi}{2}),(\frac{-\pi}{2}, \frac{\pi}{2}),(\frac{\pi}{2}, \frac{3 \pi}{2})$ में से कोई भी हो सकता है। इन अंतरालों द्वारा फलन tan' की
भिन्न-भिन्न शाखाएँ मिलती हैं। वह शाखा, जिसका परिसर   $(\frac{-\pi}{2}, \frac{\pi}{2})$  होता है, फलन tan¹ की  मुख्य शाखा कहलाती है। इस प्रकार

$$
\tan ^{-1}: \mathbf{R} \rightarrow(\frac{-\pi}{2}, \frac{\pi}{2})
$$

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$ <span style="color:blue"> $\tan ^{-1}$ Function</span> $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Graph of $\tan ^{-1}$ Function</B></Primary>

<hr>

<main>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ <span style="color:blue">Graph of $\tan ^{-1}$ Function</span>  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>$\cot ^{-1}$ Function</B></Primary>

<hr>

<main>

$\cot ^{-1}$ can be defined as a function whose domain is the $\mathbf{R}$ and range as any of the
intervals $(-\pi, 0),(0, \pi),(\pi, 2 \pi)$ etc. These intervals give different branches of the function $\cot ^{-1}$. The function with range $(0, \pi)$ is called the principal value branch of the function $\cot ^{-1}$. We thus have

$$
\cot ^{-1}: \mathbf{R} \rightarrow(0, \pi)
$$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

$\cot ^{-1}$ एक ऐसे फलन के रूप में परिभाषित हो सकता है, जिसका प्रांत R हो और परिसर, अंतरालों (-π, 0), (0, π), (π, 2) इत्यादि में से कोई भी हो। इन अंतरालों से फलन cot की भिन्न-भिन्न शाखाएँ प्राप्त होती हैं। वह शाखा, जिसका परिसर (0, π) होता है, फलन cot' की मुख्य शाखा कहलाती है। इस प्रकार

$$
\cot ^{-1}: \mathbf{R} \rightarrow(0, \pi)
$$

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  <span style="color:blue">$\cot ^{-1}$ Function</span> $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Graph of $\cot ^{-1}$ Function</B></Primary>

<hr>

<main>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ <span style="color:blue">Graph of $\cot ^{-1}$ Function</span> $\rightarrow$ Note  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Summary</B></Primary>

<hr>

<main>

<table>
  <tr>
    <th>Function</th>
    <th>Domain</th>
    <th>Range</th>
  </tr>
  <tr>
    <td>sin⁻¹</td>
    <td>[-1,1]</td>
    <td>[-π/2, π/2] </td>
  </tr>
  <tr>
    <td>cos⁻¹</td>
    <td>[-1,1]</td>
    <td>[0, π]</td>
  </tr>
  <tr>
    <td>cosec⁻¹</td>
    <td>R - (-1,1)</td>
    <td>[π/2, π/2] - {0}</td>
  </tr>
  <tr>
    <td>sec⁻¹</td>
    <td>R - (-1,1)</td>
    <td>[0, π] - {π/2}</td>
  </tr>
  <tr>
    <td>tan⁻¹</td>
    <td>R</td>
    <td>(-π/2, π/2)</td>
  </tr>
  <tr>
    <td>cot⁻¹ </td>
    <td>R</td>
    <td>(0, π)</td>
  </tr>
</table>
</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ <span style="color:blue">Note</span>  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Note</B></Primary>

<hr>

<main>

**1.** $\sin ^{-1} x$ should not be confused with $(\sin x)^{-1}$. In fact $(\sin x)^{-1}=\frac{1}{\sin x}$ and similarly for other trigonometric functions.

**2.** Whenever no branch of an inverse trigonometric functions is mentioned, we mean the principal value branch of that function.

**3.** The value of an inverse trigonometric functions which lies in the range of principal branch is called the principal value of that inverse trigonometric functions.

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

1. $\sin ^{-1} x$ से $(\sin x)^{-1}$ की भ्रांति नहीं होनी चाहिए। वास्तव में $(\sin x)^{-1}=\frac{1}{\sin x}$ और यह तथ्य
अन्य त्रिकोणमितीय फलनों के लिए भी सत्य होता है।

2. जब कभी प्रतिलोम त्रिकोणमितीय फलनों की किसी शाखा विशेष का उल्लेख न हो, तो हमारा तात्पर्य उस फलन की मुख्य शाखा से होता है।

3. किसी प्रतिलोम त्रिकोणमितीय फलन का वह मान, जो उसकी मुख्य शाखा में स्थित होता है, प्रतिलोम त्रिकोणमितीय फलन का मुख्य मान (Principal value) कहलाता है।

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ <span style="color:blue">Note</span>  $\rightarrow$ Examples</footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Example</B></Primary>

<hr>

<main>

**Example:** Find the principal value of $\sin ^{-1}(\frac{1}{\sqrt{2}})$.

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

**उदाहरण**  $\sin ^{-1}(\frac{1}{\sqrt{2}})$ मुख्य मान ज्ञात कीजिए

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ <span style="color:blue">Examples</span></footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Solution</B></Primary>

<hr>

<main>

**Solution** Let $\sin ^{-1}(\frac{1}{\sqrt{2}})=y$. Then, $\sin y=\frac{1}{\sqrt{2}}$.

We know that the range of the principal value branch of $\sin ^{-1}$ is $\frac{-\pi}{2}, \frac{\pi}{2}$ and $\sin (\frac{\pi}{4})=\frac{1}{\sqrt{2}}$. Therefore, principal value of $\sin ^{-1}(\frac{1}{\sqrt{2}})$ is $\frac{\pi}{4}$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

**हल** मान लीजिए कि  $\sin ^{-1}(\frac{1}{\sqrt{2}})=y$  अतः   $\sin y=\frac{1}{\sqrt{2}}$  हमें ज्ञात है कि sin' की मुख्य शाखा का परिसर
$\frac{-\pi}{2}, \frac{\pi}{2}$  होता और   $\sin (\frac{\pi}{4})=\frac{1}{\sqrt{2}}$  इसलिए    $\sin ^{-1}(\frac{1}{\sqrt{2}})$  का मुख्य मान  
$\frac{\pi}{4}$
</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ <span style="color:blue">Examples</span></footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Example</B></Primary>

<hr>

<main>

**Example** Find the principal value of $\cot ^{-1}(\frac{-1}{\sqrt{3}})$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

$\cot ^{-1}(\frac{-1}{\sqrt{3}})$ मुख्य मान ज्ञात कीजिए

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ <span style="color:blue">Examples</span></footer>

</div>

======

<Success style = "float:center; text-align:center; font-size: 22px;"><B>Inverse Trigonometric Functions(ITF)</B></Success>

#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Solution</B></Primary>

<hr>

<main>

**Solution**  Let $\cot ^{-1}(\frac{-1}{\sqrt{3}})=y$. Then,

$$
\cot y=\frac{-1}{\sqrt{3}}=-\cot (\frac{\pi}{3})=\cot (\pi-\frac{\pi}{3})=\cot (\frac{2 \pi}{3})
$$

We know that the range of principal value branch of $\cot ^{-1}$ is $(0, \pi)$ and $\cot (\frac{2 \pi}{3})=\frac{-1}{\sqrt{3}}$. Hence, principal value of $\cot ^{-1}(\frac{-1}{\sqrt{3}})$ is $\frac{2 \pi}{3}$

</div>
<div  style='float: right; width:49%; text-align: left;font-size:21px'>

**हल**  मान लीजिए कि   $\cot ^{-1}(\frac{-1}{\sqrt{3}})=y$

$$
\cot y=\frac{-1}{\sqrt{3}}=-\cot (\frac{\pi}{3})=\cot (\pi-\frac{\pi}{3})=\cot (\frac{2 \pi}{3})
$$

हमें ज्ञात है कि cot' की मुख्य शाखा का परिसर (0, ) होता है और   $\cot (\frac{2 \pi}{3})=\frac{-1}{\sqrt{3}}$

$\cot ^{-1}(\frac{-1}{\sqrt{3}})$  का मुख्य मान  $\frac{2 \pi}{3}$ है।

</div>

</main>

<hr>

<footer> Basic Concepts $\rightarrow$  $\sin^{-1} $ Function $\rightarrow$ Graph of $\sin^{-1} $ Function $\rightarrow$  $\cos^{-1} $ Function  $\rightarrow$ Graph of $\cos^{-1} $ Function   $\rightarrow$   $\cosec^{–1}$ Function $\rightarrow$  Graph of  $\cosec^{-1}$ Function   $\rightarrow$  $\sec ^{-1}$ Function $\rightarrow$  $\tan ^{-1}$ Function $\rightarrow$ Graph of $\tan ^{-1}$ Function  $\rightarrow$  $\cot ^{-1}$ Function $\rightarrow$ Graph of $\cot ^{-1}$ Function $\rightarrow$ Note  $\rightarrow$ <span style="color:blue">Examples</span></footer>

</div>

======

======
#### <Primary style = "float:center; text-align:center; font-size: 22px;"><B>Don't forget to:</B></Primary>

👉 Register yourself and give subject-wise mock test.

<p id="blinking-text"><span style="color:blue"><a href="https://jeetest.prutor.ai/">https://jeetest.prutor.ai/</a></span></p>

👉  Register yourself and ask any doubt regarding these video lectures or any subject doubt.

<p id="blinking-text"><span style="color:blue"><a href="https://satheeqrcode.web.app/">https://satheeqrcode.web.app/</a></span></p>

👉 Here you can find "how to videos"

<p id="blinking-text"><span style="color:blue"><a href="https://www.youtube.com/playlist?list=PL5_E7EoTpHVfPmpOjbs9rKQwdSxyOyBXE">https://www.youtube.com/playlist?list=PL5_E7EoTpHVfPmpOjbs9rKQwdSxyOyBXE</a></span></p>

👉 Also, visit <p id="blinking-text"><span style="color:blue"><a href="https://sathee.prutor.ai/">https://sathee.prutor.ai/</a></span></p>

</div>

👉 अपना रजिस्ट्रेशन करें और विषयवार मॉक टेस्ट दें।

<p id="blinking-text"><span style="color:blue"><a href="https://jeetest.prutor.ai/">https://jeetest.prutor.ai/</a></span></p>

👉   स्वयं को पंजीकृत करें और इन वीडियो व्याख्यानों के संबंध में कोई संदेह या किसी विषय पर संदेह पूछें।

<p id="blinking-text"><span style="color:blue"><a href="https://satheeqrcode.web.app/">https://satheeqrcode.web.app/</a></span></p>

👉 यहां आप "पंजीकरण कैसे करें" पा सकते हैं -

👉 इसके अलावा, <p id='blinking-text'><span style='color:blue'><a href='https://sathee.prutor.ai/'>https://sathee.prutor.ai/</a></span></p>पर भी जाएं।

</div>

======

$\quad$

### <Primary>Thank you</Primary>