Inverse Trigonometric Functions

📚 Complete Practice Guide: Inverse Trigonometric Functions – Class 12

Inverse trigonometric functions are used when we know a trigonometric ratio and want to find the corresponding angle. In Class 12 Mathematics, this chapter develops the meaning of inverse functions, principal values, domains, ranges, identities and transformations involving sin⁻¹x, cos⁻¹x and tan⁻¹x. The most important skill is to remember that inverse trigonometric functions are defined using carefully restricted principal branches. This worksheet gives repeated practice so students can build speed and accuracy.

1. What does an inverse trigonometric function mean?

For a function to have an inverse, it must be one-to-one on the chosen restricted domain. Since the ordinary sine, cosine and tangent functions repeat values, their domains are restricted before defining inverse functions. The inverse functions are written as sin⁻¹x, cos⁻¹x, and tan⁻¹x. The notation means inverse function, not reciprocal. Thus sin⁻¹x is not 1/sin x.

2. ⭐ Principal value ranges

sin⁻¹x ∈ [−π/2, π/2]

cos⁻¹x ∈ [0, π]

tan⁻¹x ∈ (−π/2, π/2)

These ranges are extremely important in evaluation questions. Whenever an answer is obtained, it must lie in the principal value range of the inverse function being used.

3. ⭐ Domains and ranges

For sin⁻¹x, the input must satisfy −1 ≤ x ≤ 1. Its output lies between −π/2 and π/2. For cos⁻¹x, again −1 ≤ x ≤ 1, while the output is from 0 to π. For tan⁻¹x, every real number is allowed as input, and the output lies strictly between −π/2 and π/2.

4. ⭐ Standard values

Students should know the common values quickly. For example, sin⁻¹(0)=0, sin⁻¹(1)=π/2, sin⁻¹(−1)=−π/2; cos⁻¹(1)=0, cos⁻¹(0)=π/2, cos⁻¹(−1)=π; and tan⁻¹(0)=0. Familiarity with π/6, π/4 and π/3 values greatly improves speed.

5. ⭐ Core identities

sin(sin⁻¹x)=x,   cos(cos⁻¹x)=x,   tan(tan⁻¹x)=x

sin⁻¹(−x)=−sin⁻¹x

tan⁻¹(−x)=−tan⁻¹x

cos⁻¹(−x)=π−cos⁻¹x

Do not apply an identity blindly: the domain and principal range determine whether a transformation is valid.

6. ⭐ Important complementary relations

sin⁻¹x + cos⁻¹x = π/2

For suitable real values, complementary relations allow a difficult-looking expression to be converted into a simpler one. Such questions are common in revision and objective practice.

7. Addition and subtraction of tan⁻¹ terms

tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy))

tan⁻¹x − tan⁻¹y = tan⁻¹((x−y)/(1+xy))

These forms require attention to the principal-value adjustment. In many school-level questions, the numerical values are selected so that the intended principal angle is easy to identify, but students should still check the range.

8. Simplification strategy

First identify the inverse functions present. Next replace known standard values, apply symmetry or complementary identities, and only then simplify. Avoid converting every expression into a calculator decimal. Exact answers involving π are normally preferred in mathematical practice.

9. Solving inverse trigonometric equations

When solving equations, first determine the permitted domain. Then use the principal value range and, where necessary, the periodic nature of the original trigonometric function. Check each candidate in the original equation. A correct-looking angle outside the principal branch may be rejected.

10. How to use this worksheet effectively

Start with 10 or 20 questions and select Mixed mode. Attempt every question before revealing the answer. Use the Check Answer button after each response. Correct answers turn green and wrong answers turn red. After checking, the Show Answer button becomes available. Increase difficulty when your accuracy is consistently high. Use Shuffle Whole Worksheet for a fresh ordering, and start a new test when you want a completely new practice session.

11. Exam-focused checklist

• Memorise the three principal ranges.
• Know domains of sin⁻¹, cos⁻¹ and tan⁻¹.
• Practise standard angles in radians.
• Remember odd/even-style identities correctly.
• Use complementary relations confidently.
• Check principal-value restrictions before finalising answers.
• Prefer exact π-based answers when appropriate.
• Practise both MCQ and written-answer formats.

12. Final revision tip

Inverse trigonometry becomes much easier when principal values are treated as the foundation rather than a final detail. Spend a few minutes reviewing the ranges and standard values before every practice session. Then solve progressively harder identity, simplification and equation questions. Regular randomized practice helps prevent students from memorising question order and encourages genuine understanding.

Note: This is an independent DigiSadhan practice tool designed for learning and revision. The provided NCERT URL is the official textbook portal; the portal itself is presented as “Textbooks PDF (I-XII).”