1. Why Inverse Trigonometric Functions Matter 🎯

Inverse trigonometric functions help you “reverse” the usual trigonometric functions.

• Trigonometric function: angle → ratio
• Inverse trigonometric function: ratio → angle

You meet them in:

• CBSE Class 12: A full chapter with definition, principal values, identities, and simple equations.
• JEE Main/Advanced: In limits, integration, solution of equations, and graph-based questions.
• NEET: Sometimes in physics (especially SHM, waves, and optics where arcsin/arctan forms appear).

So, this Set-1 is all about:

• Understanding principal value branches
• Learning standard values
• Using basic identities to simplify expressions

2. Getting the Basics Clear 🧱

2.1 What is an inverse trigonometric function?

Example: Sine function

• If sin θ = 1/2, then one of the values of θ is 30°.
• But there are infinitely many such angles (…, -330°, -30°, 30°, 150°, 390°, …).

To define a single-valued inverse, we restrict the domain of the trigonometric function so that it becomes one-one and onto, and then define its inverse on that domain.

For sine:

• Restricted domain: $[- \pi/2,\ \pi/2]$
• Range of sin on this interval: $[-1,\ 1]$

So the inverse sine function, written as sin⁻¹x or arcsin x, is defined from $[-1, 1]$ to $[- \pi/2, \pi/2]$.

In symbols:

Similarly, for other functions:

• cos⁻¹x, tan⁻¹x, cot⁻¹x, sec⁻¹x, cosec⁻¹x

Each has its own domain and principal value range.

3. Principal Value Branches – Super Important for Exams 🔑

Memorising the principal value ranges is non‑negotiable. Many CBSE, JEE, and NEET maths questions are traps based on wrong range selection.

3.1 Quick Reference Table 📋

Memory aid:

• sin⁻¹ and tan⁻¹ → range around 0 symmetric: $-\pi/2$ to $\pi/2$
• cos⁻¹ and cot⁻¹ → 0 to $\pi$

4. Standard Values You Must Know 🧠

These are like tables for trigonometry, but now for inverse trig functions.

4.1 Basic angles table

(“–” means that pair is not usually used in basic tables.)

5. Fundamental Identities – Your Shortcut Toolbox 🧰

These are “must‑remember” identities that help you simplify expressions.

5.1 Direct inverse identities

For x in the domain of the function:

And for y in the principal value range:

Note the condition:
The second set only works when y is within the principal range of the respective inverse function.

6. Classic Solved Examples (Step‑by‑Step) ✍️

Example 1: Evaluating a simple inverse trig value

Evaluate sin⁻¹(√3/2).

Step 1: Recall standard sine values
We know sin 60° = √3/2, i.e., sin(π/3) = √3/2.

Step 2: Check if angle is in principal range
Principal value range for sin⁻¹x is $[- \pi/2, \pi/2]$.
π/3 lies in this interval.

Therefore,

Example 2: Carefully applying principal value

Evaluate cos⁻¹(-1/2).

Step 1: Know which angle gives cos θ = -1/2
cos 120° = -1/2, i.e., cos(2π/3) = -1/2.

Step 2: Check principal value range
For cos⁻¹x, range is [0, π].
2π/3 lies in this range.

So,

Note: Even though cos 4π/3 = -1/2 also, we never use 4π/3 here because it lies outside [0, π].

Example 3: Composition involving tan and tan⁻¹

Simplify tan(tan⁻¹ 5).

Step 1: Identify the pattern
This is of the form tan(tan⁻¹ x).

For all real x:

So,

No need for detailed calculation.

Example 4: Composition in reverse order – more subtle

Simplify tan⁻¹(tan 4π/3).

Step 1: Identify principal value range
Range of tan⁻¹x is $-\pi/2,\ \pi/2$.

Step 2: Reduce 4π/3 to this interval

4π/3 is in the 3rd quadrant.

We know tan(θ) has period π.
So subtract π:

And tan(4π/3) = tan(π/3).

Now π/3 is about 1.047, which is not in $-\pi/2,\ \pi/2$ (since π/2 ≈ 1.57, actually π/3 is in this interval – so let’s reason directly).

Better route:

π/3 is indeed between -π/2 and π/2, so

But tan(4π/3) = tan(π/3), so:

Hence,

Key learning:
For tan⁻¹(tan θ), always convert θ to an equivalent angle in $-\pi/2,\ \pi/2$.

7. Special Identities Connecting Different Inverses 🔄

Some very popular identities often used in JEE/boards:

1. Relation between sin⁻¹ and cos⁻¹

2. Relation between tan⁻¹ and cot⁻¹

3. Relation between tan⁻¹ and cot⁻¹ (alternate form)

and

Quick Revision Box 📦

• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• Always think: “Is my final angle in the principal range?”
• For sin⁻¹x: angle in $[- \pi/2, \pi/2]$
• For cos⁻¹x: angle in [0, π]
• For tan⁻¹x: angle in $-\pi/2, \pi/2$

8. Mixed Example Using Identities 🧮

Example 5: Evaluate sin⁻¹(3/5) + cos⁻¹(3/5)

We directly use the identity:

So, taking x = 3/5:

No calculation required.

Example 6: Simplify tan⁻¹(1/3) + tan⁻¹(2/5)

We can use the formula:

provided that the product ab < 1 and the sum lies in the principal range.

Here, a = 1/3, b = 2/5.

Compute:

• a + b = 1/3 + 2/5 = (5 + 6)/15 = 11/15
• ab = (1/3) × (2/5) = 2/15 < 1, so formula is valid.

So,

Now,

1 - 2/15 = 13/15

Thus:

Therefore,

Since 11/13 is positive and tan⁻¹x of a positive number lies in (0, π/2), which is in principal range, this is our final answer.

9. Typical Mistakes Students Make (and How to Avoid Them) 🚫

10. Strategy Tips for CBSE, JEE & NEET 🧠⚙️

For CBSE Class 12

• Focus on:
  • Definitions, domains, ranges
  • Simple identities and compositions
  • Proving small identities using basic trigonometry
• Practice NCERT Examples + Exercise questions thoroughly.
• In board exams, inverse trig questions are usually straightforward: evaluate, simplify, or prove.

For JEE Main / Advanced

• Expect:
  • Expressions like tan⁻¹x + tan⁻¹y
  • Transformations using identities
  • Links with algebra (substitutions like x = tan θ)
• Always think:
  • Can I convert this expression into a single inverse trig term?
  • Is there a smart substitution using right triangle geometry?

For NEET Aspirants

• Direct inverse trig is rare in NEET maths, but:
  • It shows up in physics, especially when solving for angles from ratios.
  • Being comfortable with principal values helps avoid conceptual errors.

11. Visualising Inverse Trigonometric Functions (In Your Mind) 🧠🎨

You don’t need to draw perfect graphs in the exam, but a mental picture helps.

sin⁻¹x graph

• Domain: -1 to 1
• Range: -π/2 to π/2
• Increasing curve passing through origin (0, 0).

Imagine taking the graph of y = sin x on $[- \pi/2,\ \pi/2]$ and reflecting it in the line y = x. That reflection is y = sin⁻¹x.

tan⁻¹x graph

• Domain: all real numbers
• Range: -π/2 to π/2
• Passes through (0, 0)
• Has horizontal asymptotes at y = π/2 and y = -π/2

This helps you guess the sign and approximate size of angles when x is large or small.

12. Mini Practice Drill 💪

Try these without looking at the solutions first:

1. Evaluate cos⁻¹(√3/2).
2. Find the value of sin⁻¹(1) + cos⁻¹(1).
3. Simplify tan⁻¹(2) + tan⁻¹(3).
4. Find the principal value of sin⁻¹(sin 7π/6).
5. If sin⁻¹x + cos⁻¹x = α, what is α?

Quick Answers (for self-check)

1. cos⁻¹(√3/2) = π/6
2. sin⁻¹(1) = π/2, cos⁻¹(1) = 0 → sum = π/2
3. Use formula: tan⁻¹a + tan⁻¹b = tan⁻¹((a + b)/(1 - ab))
   • Here a = 2, b = 3 → (a + b) = 5, ab = 6
   • So 1 - ab = -5
   • Expression = tan⁻¹(5 / -5) = tan⁻¹(-1)
   • tan⁻¹(-1) = -π/4 (since -1 lies in principal range)
4. sin 7π/6 = sin(π + π/6) = -1/2
   • sin⁻¹(-1/2) = -π/6 (principal value)
5. Always α = π/2, since sin⁻¹x + cos⁻¹x = π/2

13. How to Tackle “Set‑1” Type Questions Efficiently ⏱️

When you see objective questions (like in a quiz or MCQ test) on this topic:

1. Scan for pattern

   • Is it of type sin⁻¹x + cos⁻¹x?
   • Or tan⁻¹a + tan⁻¹b?
   • Or tan⁻¹(tan θ)?

2. Immediately recall identity

   • Don’t re‑derive in the exam; directly apply.

3. Check range at the end

   • If your angle is outside the principal range, bring it inside using periodicity.

4. Use right triangle logic for simple ratios

   • For sin⁻¹(3/5), think of a right triangle with opposite = 3, hypotenuse = 5.

This habit will save precious minutes in both boards and entrance exams.

14. Ready to Test Yourself? 🚀

Now that you’ve revised the core ideas, identities, and typical questions from Inverse Trigonometric Functions Set‑1, it’s the perfect time to evaluate your understanding with a focused practice quiz.