📘 Detailed Summary: Inverse Trigonometric Functions (Class 12 NCERT)

The chapter Inverse Trigonometric Functions introduces you to the reverse process of trigonometric functions — meaning, instead of finding the trigonometric value of an angle, you find the angle whose trigonometric value is known.

But since trigonometric functions are periodic (they repeat), they’re not one-one and hence don’t have inverses on their full domain.
So the chapter focuses on restricting their domains to make them one-one, and then defining their inverses.

1️⃣ Why Do We Need Restricted Domains?

Trigonometric functions like sin, cos, tan keep repeating, so they’re not one-one on ℝ.
To define an inverse, each function must first be made bijective.

So NCERT chooses standard restricted domains:

• sin x → [−π2,π2][- \frac{\pi}{2}, \frac{\pi}{2}][−2π​,2π​]

• cos x → $[0,\pi ]$

• tan x → (−π2,π2)(- \frac{\pi}{2}, \frac{\pi}{2})(−2π​,2π​)

• cot x → $(0,\pi )$

• sec x → [0,π]∖{π2}[0, \pi] \setminus \{\frac{\pi}{2}\}[0,π]∖{2π​}

• cosec x → [−π2,π2]∖{0}[- \frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}[−2π​,2π​]∖{0}

On these intervals, the functions become one–one and onto, so inverses can be defined properly.

2️⃣ Definition of Inverse Trigonometric Functions

The inverse functions are written as:

• sin⁡−1x\sin^{-1}xsin−1x

• cos⁡−1x\cos^{-1}xcos−1x

• tan⁡−1x\tan^{-1}xtan−1x

• cot⁡−1x\cot^{-1}xcot−1x

• sec⁡−1x\sec^{-1}xsec−1x

• csc⁡−1x\csc^{-1}xcsc−1x

Each of these takes a value and returns the principal angle that lies in the restricted range.

For example:

sin⁡−1(x)=θsuch that sin⁡θ=x and θ∈[−π2,π2]\sin^{-1}(x) = \theta \quad \text{such that } \sin \theta = x \text{ and } \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]sin−1(x)=θsuch that sinθ=x and θ∈[−2π​,2π​]

3️⃣ Domains and Ranges of Inverse Trigonometric Functions

(i) sin⁻¹ x

• Domain: $[-1,1]$

• Range: [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right][−2π​,2π​]

(ii) cos⁻¹ x

• Domain: $[-1,1]$

• Range: $[0,\pi ]$

(iii) tan⁻¹ x

• Domain: ℝ

• Range: (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)(−2π​,2π​)

(iv) cot⁻¹ x

• Domain: ℝ

• Range: $(0,\pi )$

(v) sec⁻¹ x

• Domain: $(-\infty ,-1]\cup [1,\infty )$

• Range: [0,π]∖{π2}[0, \pi] \setminus \{\frac{\pi}{2}\}[0,π]∖{2π​}

(vi) cosec⁻¹ x

• Domain: $(-\infty ,-1]\cup [1,\infty )$

• Range: [−π2,π2]∖{0}\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \setminus \{0\}[−2π​,2π​]∖{0}

These ranges are the principal values.

4️⃣ Important Properties

a) Basic Inverse Identities

sin⁡(sin⁡−1x)=x\sin(\sin^{-1}x) = xsin(sin−1x)=x cos⁡(cos⁡−1x)=x\cos(\cos^{-1}x) = xcos(cos−1x)=x tan⁡(tan⁡−1x)=x\tan(\tan^{-1}x) = xtan(tan−1x)=x

But the reverse identities depend on the quadrant:

sin⁡−1(sin⁡x)=xonly if x∈[−π2,π2]\sin^{-1}(\sin x) = x \quad \text{only if } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]sin−1(sinx)=xonly if x∈[−2π​,2π​] cos⁡−1(cos⁡x)=xonly if x∈[0,π]\cos^{-1}(\cos x) = x \quad \text{only if } x \in [0, \pi]cos−1(cosx)=xonly if x∈[0,π]

5️⃣ Relationships Between Inverse Functions

Some extremely helpful identities:

Complementary Angle Relations

sin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}sin−1x+cos−1x=2π​ tan⁡−1x+cot⁡−1x=π2\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}tan−1x+cot−1x=2π​

Negative Angle Relations

sin⁡−1(−x)=−sin⁡−1(x)\sin^{-1}(-x) = -\sin^{-1}(x)sin−1(−x)=−sin−1(x) tan⁡−1(−x)=−tan⁡−1(x)\tan^{-1}(-x) = -\tan^{-1}(x)tan−1(−x)=−tan−1(x) cos⁡−1(−x)=π−cos⁡−1(x)\cos^{-1}(-x) = \pi – \cos^{-1}(x)cos−1(−x)=π−cos−1(x)

6️⃣ Miscellaneous Important Results

tan⁡−1x+tan⁡−1y={tan⁡−1(x+y1−xy),xy<1tan⁡−1(x+y1−xy)+π,x>0,y>0,xy>1\tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\left(\frac{x + y}{1 – xy}\right), & xy < 1 \\ \tan^{-1}\left(\frac{x + y}{1 – xy}\right) + \pi, & x > 0, y > 0, xy > 1 \end{cases}tan−1x+tan−1y=⎩⎨⎧​tan−1(1−xyx+y​),tan−1(1−xyx+y​)+π,​xy<1x>0,y>0,xy>1​ tan⁡−1x−tan⁡−1y=tan⁡−1(x−y1+xy),1+xy≠0\tan^{-1}x – \tan^{-1}y = \tan^{-1}\left(\frac{x – y}{1 + xy}\right), \quad 1 + xy \neq 0tan−1x−tan−1y=tan−1(1+xyx−y​),1+xy=0

These are super useful in board questions.

7️⃣ Graphical Behaviour

The chapter also introduces graphs of inverse functions:

• Increasing/decreasing nature

• Symmetry about line $y=x$

• Graph of y=sin⁡−1xy = \sin^{-1}xy=sin−1x and y=cos⁡−1xy = \cos^{-1}xy=cos−1x compared with sin and cos

It gives you visual insight into why we restrict domains.

8️⃣ Applications

Inverse trigonometric functions are essential in:

• Calculus (especially integration)

• Geometry

• Coordinate geometry

• Solving equations

• Modelling angles where direct trig functions don’t work cleanly