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CBSE · Class 12 · Mathematics

Properties of Inverse Trigonometric Functions (Ch 2: 2.3)

3 concepts · 24 practice questions with answers · Free to read

Inverse of a function and the function of an inverse

From daily life

Wrapping a gift and then unwrapping returns the gift. But if the wrapping is too big, unwrapping gives only the part you are allowed to see.

Simple definition

sin(sin⁻¹ x) = x for x in [−1, 1]. But sin⁻¹(sin x) = x only if x lies in [−π/2, π/2]. For other x, first find the equal sine in the principal range. Similarly for cos⁻¹(cos x) on [0, π] and tan⁻¹(tan x) on (−π/2, π/2).

Formula: sin(sin⁻¹ x) = x, sin⁻¹(sin x) = x for x in [−π/2, π/2]

Rule 1: Value already inside the range — The two functions cancel.

  • Easy: sin(sin⁻¹(3/5)) = 3/5.
  • Medium: cos⁻¹(cos(π/7)) = π/7 and sin⁻¹(sin(π/4)) = π/4.
  • Exam-type: tan⁻¹(tan(π/9)) = π/9.

Rule 2: Value outside the range — Reduce to an equal value inside the range.

  • Easy: sin⁻¹(sin(3π/4)) = sin⁻¹(√2/2) = π/4.
  • Medium: cos⁻¹(cos(7π/6)) = cos⁻¹(−√3/2) = 5π/6.
  • Exam-type: sin⁻¹(sin 3) = π − 3, because sin 3 = sin(π − 3) and π − 3 is about 0.14, inside the range.
Remember

Always check which range the angle is in before cancelling.

Common mistake

Cancelling sin⁻¹(sin x) to x for every x.

Memory clue

Ask "is the angle in the principal box?" before cancelling.

Practice questions

Easy
  1. 1Find sin(sin⁻¹(2/5)).
  2. 2Find cos⁻¹(cos(π/7)).
  3. 3Find sin⁻¹(sin(π/4)).
Medium
  1. 4Find sin⁻¹(sin(5π/6)).
  2. 5Find cos⁻¹(cos(7π/6)).
  3. 6Find tan⁻¹(tan(2π/3)).
Challenging
  1. 7Find sin⁻¹(sin 3).
  2. 8Find cos⁻¹(cos 4).
Show answers (try them yourself first)
  1. 12/5
  2. 2π/7
  3. 3π/4
  4. 4sin(5π/6) = 1/2, so the answer is π/6
  5. 5cos(7π/6) = −√3/2, so the answer is 5π/6
  6. 6tan(2π/3) = −√3, so the answer is −π/3
  7. 7π − 3
  8. 8cos 4 = cos(2π − 4) and 2π − 4 is about 2.28, inside [0, π]; the answer is 2π − 4

Negative arguments and complementary pairs

From daily life

Reflect an angle in a mirror and its sine changes sign; add an angle to its complement and you get a right angle. These simple facts become identities.

Simple definition

Sin⁻¹ and tan⁻¹ are odd: sin⁻¹(−x) = −sin⁻¹ x and tan⁻¹(−x) = −tan⁻¹ x. For cos⁻¹ and cot⁻¹: cos⁻¹(−x) = π − cos⁻¹ x, cot⁻¹(−x) = π − cot⁻¹ x. Also sin⁻¹ x + cos⁻¹ x = π/2 and tan⁻¹ x + cot⁻¹ x = π/2.

Formula: sin⁻¹ x + cos⁻¹ x = π/2, tan⁻¹ x + cot⁻¹ x = π/2, cos⁻¹(−x) = π − cos⁻¹ x

Rule 1: Use the complementary pair — Add inverse sine and inverse cosine of the same number.

  • Easy: sin⁻¹(0.3) + cos⁻¹(0.3) = π/2.
  • Medium: If sin⁻¹ x = π/5 then cos⁻¹ x = π/2 − π/5 = 3π/10.
  • Exam-type: sin⁻¹(1/3) + cos⁻¹(1/3) + tan⁻¹ 1 = π/2 + π/4 = 3π/4.

Rule 2: Remove the minus sign — Use the rules for negative arguments.

  • Easy: cos⁻¹(−1/2) = π − π/3 = 2π/3.
  • Medium: tan⁻¹(−√3) = −π/3.
  • Exam-type: sin⁻¹(−1/2) + cos⁻¹(−1/2) = −π/6 + 2π/3 = π/2.
Remember

The pair rules need the same number x in both functions.

Common mistake

Using cos⁻¹(−x) = −cos⁻¹ x; for cos⁻¹ you must use π − cos⁻¹ x.

Memory clue

Sin and tan are odd; cos and cot are "π minus".

Practice questions

Easy
  1. 1Find sin⁻¹(0.6) + cos⁻¹(0.6).
  2. 2Find cos⁻¹(−1/2).
  3. 3Find tan⁻¹ 5 + cot⁻¹ 5.
Medium
  1. 4If sin⁻¹ x = π/10, find cos⁻¹ x.
  2. 5Find cot⁻¹(−1).
  3. 6Write cosec⁻¹ 3 as an inverse sine.
Challenging
  1. 7Find sec⁻¹(−2) + cosec⁻¹(−2).
  2. 8Find cos⁻¹(−√3/2) + sin⁻¹(√3/2).
Show answers (try them yourself first)
  1. 1π/2
  2. 2π − π/3 = 2π/3
  3. 3π/2
  4. 4π/2 − π/10 = 2π/5
  5. 5π − π/4 = 3π/4
  6. 6sin⁻¹(1/3)
  7. 72π/3 + (−π/6) = π/2
  8. 85π/6 + π/3 = 7π/6

Sum, difference and double of tan⁻¹

From daily life

Two slopes combine into a new angle when two ramps are joined. The tangent addition rule gives the slope of the joined ramp.

Simple definition

For xy < 1, tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y) / (1 − xy)). For xy > −1, tan⁻¹ x − tan⁻¹ y = tan⁻¹((x − y) / (1 + xy)). For |x| < 1, 2 tan⁻¹ x = tan⁻¹(2x / (1 − x²)).

Formula: tan⁻¹ x + tan⁻¹ y = tan⁻¹ ((x + y) / (1 − xy)), 2 tan⁻¹ x = tan⁻¹ (2x / (1 − x²))

Rule 1: Combine two inverse tangents — Check the condition on xy first.

  • Easy: tan⁻¹(1/2) + tan⁻¹(1/3) = tan⁻¹((5/6) / (5/6)) = tan⁻¹ 1 = π/4.
  • Medium: tan⁻¹(1/2) − tan⁻¹(1/3) = tan⁻¹((1/6) / (7/6)) = tan⁻¹(1/7).
  • Exam-type: tan⁻¹(1/4) + tan⁻¹(3/5) = tan⁻¹((17/20) / (17/20)) = π/4.

Rule 2: Double the angle — Put the number in 2x / (1 − x²).

  • Easy: 2 tan⁻¹(1/3) = tan⁻¹((2/3) / (8/9)) = tan⁻¹(3/4).
  • Medium: 2 tan⁻¹(1/2) = tan⁻¹(1 / (3/4)) = tan⁻¹(4/3).
  • Exam-type: cos(2 tan⁻¹(1/3)) = (1 − 1/9) / (1 + 1/9) = 4/5.
Remember

If xy is more than 1 in the sum formula, add π (for positive x, y) because the angle is beyond π/2.

Common mistake

Applying the sum formula when xy ≥ 1 without correcting by π.

Memory clue

"Sum over one minus product" — the tangent addition rule in disguise.

Practice questions

Easy
  1. 1Find tan⁻¹(1/5) + tan⁻¹(2/3).
  2. 2Find tan⁻¹(2/3) − tan⁻¹(1/5).
  3. 3Write 2 tan⁻¹(1/4) as a single tan⁻¹.
Medium
  1. 4Find cos(2 tan⁻¹(1/2)).
  2. 5Find tan⁻¹(1/7) + tan⁻¹(1/8).
  3. 6Find tan⁻¹ 3 − tan⁻¹ 2.
Challenging
  1. 7Find sin(2 tan⁻¹(1/3)).
  2. 8Find tan⁻¹ 2 + tan⁻¹ 3, taking care of the condition xy < 1.
Show answers (try them yourself first)
  1. 1(13/15) / (13/15) = 1, so π/4
  2. 2tan⁻¹((7/15) / (17/15)) = tan⁻¹(7/17)
  3. 3tan⁻¹((1/2) / (15/16)) = tan⁻¹(8/15)
  4. 4(1 − 1/4) / (1 + 1/4) = 3/5
  5. 5tan⁻¹((15/56) / (55/56)) = tan⁻¹(3/11)
  6. 6tan⁻¹(1 / 7)
  7. 72x / (1 + x²) = (2/3) / (10/9) = 3/5
  8. 8xy = 6 is more than 1, so the sum is π + tan⁻¹(5 / (1 − 6)) = π + tan⁻¹(−1) = 3π/4