📌 1. Introduction

We know:$$\sin x, \cos x, \tan x$$sinx,cosx,tanx

But what if we want to find the angle when the value is given?

👉 Example:$$\sin x = \frac{1}{2}$$sinx=21​

Then:$$x = \sin^{-1}\left(\frac{1}{2}\right)$$x=sin−1(21​)

This leads to inverse trigonometric functions.

---

📖 2. Definition

If:$$y = \sin x$$y=sinx

Then:$$x = \sin^{-1} y$$x=sin−1y

---

📌 Important Note:

Inverse trigonometric functions are also called arc functions.

---

📊 3. Domain and Range

Since trigonometric functions are not one-one, we restrict their domains to make them invertible.

---

🔹 1. Sine Inverse

$$y = \sin^{-1} x$$y=sin−1x

---

$y = \sin^{-1} x$y=sin−1x

• Domain: $[-1, 1]$[−1,1]
• Range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$[−2π​,2π​]

---

🔹 2. Cosine Inverse

$$y = \cos^{-1} x$$y=cos−1x

---

$y = \cos^{-1} x$y=cos−1x

• Domain: $[-1, 1]$[−1,1]
• Range: $[0, \pi]$[0,π]

---

🔹 3. Tangent Inverse

$$y = \tan^{-1} x$$y=tan−1x

---

$y = \tan^{-1} x$y=tan−1x

• Domain: $(-\infty, \infty)$(−∞,∞)
• Range: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$(−2π​,2π​)

---

🔹 4. Cot Inverse

$$y = \cot^{-1} x$$y=cot−1x

• Domain: $(-\infty, \infty)$(−∞,∞)
• Range: $(0, \pi)$(0,π)

---

🔹 5. Sec Inverse

$$y = \sec^{-1} x$$y=sec−1x

• Domain: $(-\infty, -1] \cup [1, \infty)$(−∞,−1]∪[1,∞)
• Range: $[0, \pi] \setminus \{\frac{\pi}{2}\}$[0,π]∖{2π​}

---

🔹 6. Cosec Inverse

$$y = \csc^{-1} x$$y=csc−1x

• Domain: $(-\infty, -1] \cup [1, \infty)$(−∞,−1]∪[1,∞)
• Range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \setminus \{0\}$[−2π​,2π​]∖{0}

---

📌 4. Principal Value Branch

Since trigonometric functions are periodic, we restrict their domain to define inverse functions uniquely.

👉 This restricted value is called the principal value.

---

📐 5. Important Identities

🔹 Basic Identities

$$\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$$sin−1x+cos−1x=2π​

---

$\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$sin−1x+cos−1x=2π​

---

$$\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$$tan−1x+cot−1x=2π​

---

$$\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}$$sec−1x+csc−1x=2π​

---

🔹 Negative Property

$$\sin^{-1}(-x) = -\sin^{-1}(x)$$sin−1(−x)=−sin−1(x)

---

$$\tan^{-1}(-x) = -\tan^{-1}(x)$$tan−1(−x)=−tan−1(x)

---

$$\cos^{-1}(-x) = \pi – \cos^{-1}(x)$$cos−1(−x)=π−cos−1(x)

---

📊 6. Graphs of Inverse Functions

🔹 Sine Inverse Graph

• Increasing function
• Passes through (0,0)

🔹 Cosine Inverse Graph

• Decreasing function

🔹 Tangent Inverse Graph

• Increasing
• Horizontal asymptotes

---

🔁 7. Relation Between Functions

$$\sin(\sin^{-1} x) = x$$sin(sin−1x)=x

---

$$\cos(\cos^{-1} x) = x$$cos(cos−1x)=x

---

$$\tan(\tan^{-1} x) = x$$tan(tan−1x)=x

---

⚠️ Important:

$$\sin^{-1}(\sin x) \neq x \text{ always}$$sin−1(sinx)=x always

It depends on the range.

---

📌 8. Properties and Results

1. Composite Forms

$$\sin^{-1} x + \sin^{-1} y = \sin^{-1}(x\sqrt{1-y^2} + y\sqrt{1-x^2})$$sin−1x+sin−1y=sin−1(x1−y2​+y1−x2​)

---

2. Special Values

• $\sin^{-1}(0) = 0$sin−1(0)=0
• $\sin^{-1}(1) = \frac{\pi}{2}$sin−1(1)=2π​
• $\cos^{-1}(1) = 0$cos−1(1)=0

---

🧩 9. Applications

• Solving trigonometric equations
• Calculus (integration & differentiation)
• Physics (angles, waves)
• Engineering problems

---

❗ Common Mistakes

• Ignoring principal value
• Confusing domain and range
• Writing wrong identities
• Assuming inverse cancels always

---

🧠 Exam Tips

• Learn ranges properly
• Practice identity-based questions
• Solve NCERT examples
• Focus on principal values

---

📚 Practice Questions

1. Find value of inverse trig expressions
2. Prove identities
3. Solve equations
4. Evaluate composite functions

---

🎯 Conclusion

Inverse Trigonometric Functions are essential for understanding advanced calculus. Mastering domains, ranges, identities, and principal values is key to solving complex problems efficiently.