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CAPF AC Inverse Trig Functions

Study Material — 1 PYQs (2019–2019) · Concept Notes · Shortcuts

CAPF AC Inverse Trig Functions is a frequently tested subtopic — 1 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.

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2019–2019
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Previous Year Questions

CAPF AC Inverse Trig Functions — Past Exam Questions

1 questions from actual CAPF AC papers · all shown free · click option to reveal solution

Exam Q 12019Previous Year Pattern

The value of sin⁻¹(sin(5π/6)) is:

Concept Notes

Inverse Trig Functions— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

Inverse trigonometric functions are the reverse operations of normal trigonometric functions. Think of them as 'undo' buttons. If sin 30° = 1/2, then sin⁻¹(1/2) = 30°. These functions help us find angles when we know the trigonometric ratios. The six inverse trigonometric functions are: sin⁻¹x, cos⁻¹x, tan⁻¹x, cosec⁻¹x, sec⁻¹x, and cot⁻¹x. The notation sin⁻¹x is also written as arcsin x.

2.Key Rules

Core rules you must know cold

Each inverse function has a specific domain and range. For sin⁻¹x: domain is [-1,1] and range is [-π/2, π/2]. For cos⁻¹x: domain is [-1,1] and range is [0, π]. For tan⁻¹x: domain is all real numbers and range is (-π/2, π/2).

3.Formula Block

Memorise — at least one formula appears in every paper
sin⁻¹x + cos⁻¹x = π/2
tan⁻¹x + cot⁻¹x = π/2
sec⁻¹x + cosec⁻¹x = π/2
sin⁻¹(-x) = -sin⁻¹x
cos⁻¹(-x) = π - cos⁻¹x
tan⁻¹(-x) = -tan⁻¹x
tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] when xy < 1

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA papers frequently test inverse trigonometric values of standard angles, domain-range problems, and simplification of expressions. Questions often involve finding exact values like sin⁻¹(√3/2) or solving equations with inverse functions.

5.Shortcuts

Use these to save 30–60 seconds per question

For quick calculation, memorize these standard values: sin⁻¹(1/2) = 30°, sin⁻¹(√2/2) = 45°, sin⁻¹(√3/2) = 60°, cos⁻¹(1/2) = 60°, tan⁻¹(1) = 45°, tan⁻¹(√3) = 60°.

6.Worked Example

Solve this step-by-step before moving on
1
Step 1

Find sin⁻¹(1/2). We know sin 30° = 1/2, so sin⁻¹(1/2) = 30° = π/6

2
Step 2

Find cos⁻¹(1/2). We know cos 60° = 1/2, so cos⁻¹(1/2) = 60° = π/3

3
Step 3

Add the results: π/6 + π/3 = π/6 + 2π/6 = 3π/6 = π/2 Answer: π/2 Worked Example 2: Simplify: tan⁻¹(1/2) + tan⁻¹(1/3)

1
Step 1

Use the formula tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)]

2
Step 2

Here x = 1/2, y = 1/3

3
Step 3

x + y = 1/2 + 1/3 = 3/6 + 2/6 = 5/6

4
Step 4

xy = (1/2)(1/3) = 1/6

5
Step 5

1 - xy = 1 - 1/6 = 5/6

6
Step 6

(x+y)/(1-xy) = (5/6)/(5/6) = 1

7
Step 7

tan⁻¹(1) = π/4 = 45° Answer: π/4 Most

7.Exam Traps

Common mistakes students make — avoid these

Students confuse sin⁻¹x with (sin x)⁻¹ = 1/sin x. Remember: sin⁻¹x means inverse sine function, NOT 1/sin x. The -1 is not an exponent here.

This confusion leads to wrong answers in 40% of student attempts. Always read the notation carefully and understand that sin⁻¹x asks 'which angle has sine equal to x?'

Key Points to Remember

  • sin⁻¹x means inverse sine, not 1/sin x - this is the most tested concept
  • Domain of sin⁻¹x and cos⁻¹x is [-1,1], domain of tan⁻¹x is all real numbers
  • Range of sin⁻¹x is [-π/2, π/2], range of cos⁻¹x is [0, π]
  • Quick formula: sin⁻¹x + cos⁻¹x = π/2 for all x in [-1,1]
  • Shortcut: tan⁻¹x + cot⁻¹x = π/2 for all positive x
  • Standard value: sin⁻¹(1/2) = π/6 = 30°, memorize all standard angles
  • Addition formula: tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] when xy < 1
  • Odd function property: sin⁻¹(-x) = -sin⁻¹x, tan⁻¹(-x) = -tan⁻¹x
  • Even function exception: cos⁻¹(-x) = π - cos⁻¹x (not odd function)
  • Range of tan⁻¹x is (-π/2, π/2), never reaches ±π/2

Exam-Specific Tips

  • sin⁻¹(√3/2) = π/3 = 60°
  • cos⁻¹(0) = π/2 = 90°
  • tan⁻¹(1/√3) = π/6 = 30°
  • sin⁻¹(1) = π/2 and sin⁻¹(-1) = -π/2
  • cos⁻¹(1) = 0 and cos⁻¹(-1) = π
  • tan⁻¹(√3) = π/3 = 60°
  • The range of sec⁻¹x is [0,π] - {π/2}
  • For tan⁻¹ addition formula, condition xy < 1 must be satisfied
Practice MCQs

Inverse Trig Functions — Practice Questions

37graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 37

All MCQs →
Practice 1easy

If tan⁻¹(x) + tan⁻¹(y) = π/4 and xy < 1, then x + y equals:

Practice 2easy

Find the value of sin⁻¹(sin(7π/6)).

Practice 3easy

If tan⁻¹(x) + tan⁻¹(y) = π/4, where x, y > 0, then xy + x + y equals:

Practice 4easy

The value of cos⁻¹(cos(−2π/3)) is:

Practice 5easy

If sin⁻¹(a) + cos⁻¹(a) = π/2 for a ∈ [−1, 1], then a equals:

Practice 6easy

If sin⁻¹(a) + cos⁻¹(a) = π/2 for a ∈ [0, 1], then which of the following is true?

Practice 7easy

Find the value of tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3).

Practice 8easy

If sin⁻¹(x) + sin⁻¹(y) = π/2, where x, y ∈ [0, 1], then which of the following is correct?

Practice 9easy

The value of tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) is:

Practice 10easy

If cos⁻¹(x) − sin⁻¹(x) = π/6, where x ∈ [0, 1], then x equals:

Practice 11easy

Find the value of tan(sin⁻¹(3/5)).

Practice 12easy

If tan⁻¹(x) + tan⁻¹(y) = π/4, where x, y > 0, then find the value of x + y + xy.

Practice 13easy

Evaluate cos⁻¹(cos(2π/3)).

Practice 14easy

The principal value of sin⁻¹(sin(7π/6)) is:

Practice 15medium

If tan⁻¹(x) + tan⁻¹(y) = π/4 and xy < 1, then x + y is equal to:

Practice 16medium

The value of tan(sin⁻¹(5/13)) is:

Practice 17medium

If tan⁻¹(a) + tan⁻¹(b) = π/4, where a, b > 0, then the relationship between a and b is:

Practice 18medium

If cos⁻¹(x) − sin⁻¹(x) = π/6, then x equals:

Practice 19medium

If sin⁻¹(x) + cos⁻¹(x) = π/2 for all x ∈ [-1, 1], then the value of sin⁻¹(3/5) + cos⁻¹(3/5) is:

Practice 20medium

If tan⁻¹(x) + tan⁻¹(y) = π/4, where x, y > 0, then the value of x + y + xy is:

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60-Second Revision — Inverse Trig Functions

  • Remember: sin⁻¹x ≠ 1/sin x, it means 'angle whose sine is x'
  • Formula: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2
  • Memorize: sin⁻¹(1/2) = 30°, sin⁻¹(√3/2) = 60°, cos⁻¹(1/2) = 60°
  • Trap: Check domain before solving - sin⁻¹ and cos⁻¹ only work for [-1,1]
  • Shortcut: Use complementary angle relationships to save calculation time
  • Pattern: Most NDA questions test standard angle values and basic properties
  • Quick check: Verify your answer lies within the correct range for each function
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