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CBSE Class 12 · Mathematics

Inverse Trigonometric Functions

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Inverse Trigonometric Functions” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Basic Concepts

Basic concepts: y = sin⁻¹x means sin y = x with y in the taught principal range (usually [−π/2, π/2]). Inverse is a range-pick, not a reciprocal. sin⁻¹(1/2)=π/6, not 2/sin. Domain [−1,1] for sin⁻¹ as taught.

Writing sin⁻¹x as 1/sin x is the leftover this heading refuses.

Figure. y = sin^{-1} x is drawn on its principal branch: domain [-1, 1], range [-π/2, π/2]. The curve passes through the origin and ends at the marked corners.

How it works

  1. Read y=sin⁻¹x as sin y=x in the principal rangeThe concept.
  2. Keep domain [−1,1] for sin⁻¹The gate.
  3. Refuse 1/sin as this symbolInverse ≠ reciprocal.

2Properties of Inverse Trigonometric Functions

Properties (school): sin⁻¹x + cos⁻¹x = π/2 on [−1,1] as taught; similar partner-sums for tan if listed. A property is an identity on a stated domain, not a slogan. One partner-sum is enough to start.

Using the sum outside [−1,1] is a domain-skip.

Figure. At x = 1/2 the two inverse values are 30° and 60°. They add to 90° = π/2, which is the identity on [-1, 1].

How it works

  1. Name the partner-sum and its domainThe property.
  2. Keep the range that made it trueHonest.
  3. Refuse a reciprocal rewriteStill inverse.

Partner sum

sin⁻¹(1/2) + cos⁻¹(1/2).

  • sin⁻¹(1/2)π/6
  • cos⁻¹(1/2)π/3
  • Sumπ/2

Pro tip. The taught partner-sum on [−1,1].

3A definition is a test you can run

Inverse-trig-word is a test: a principal-range pick or a partner-sum. If you only say “arc”, you have a heading.

A calculator-sticker is not the test.

Figure. The definition is a test: sin^{-1} is only applied to x in [-1, 1], and the output is forced into [-π/2, π/2]. x = 2 fails the domain test.

How it works

  1. Name range or partner-sumThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

4Name the given before the unknown

The given is a value in [−1,1] or a partner. The unknown is the principal angle. Copy 1/2 before you output 5π/6 for sin⁻¹.

sin⁻¹(1/2)=5π/6 steals a non-principal sine-zero.

Figure. Given opposite 3 and hypotenuse 5 on a 3-4-5 right triangle. Name those first; the unknown is θ = sin^{-1}(3/5). The right angle is at C, so the drawn legs are square.

How it works

  1. Copy the valueThe given.
  2. Name the principal rangeThe unknown’s home.
  3. Then read the angleGiven first.

5One worked case is enough at this class

One worked case is enough: tan⁻¹1 = π/4. Do not stack ten more arctans. The case teaches range-pick.

A poster of twenty inverse values is not more science.

Figure. One special value: opposite 1, hypotenuse 2, so θ = 30° = π/6. That is sin^{-1}(1/2). Close this case before opening a second identity.

How it works

  1. Take one standard valueThe case.
  2. Land in the taught rangeThe teach.
  3. Stop — one caseThis class.

6A miss: swapping the school name for the picture

A miss: swapping the school name for a picture. “Principal value” is the range-pick — a unit-circle glow is a setting.

A pretty arc without a range-word is a mute picture.

Figure. sin(2π/3) = sin(π/3), but 2π/3 is outside the principal range. The inverse returns π/3, not the input angle. Naming the input as “the answer” is the miss.

How it works

  1. Keep principal range as the nameThe science.
  2. Use a circle only as a settingHonest.
  3. Refuse a swap of name for glowThe miss named.

7Check by the opposite action or the opposite test

Check by the opposite: apply sin (or the matching fn) to your angle and recover the input, and confirm the angle sat in range. If sin(y)≠x or y is outside, the inverse-read failed.

A calculator dump with no range-check is a weak check.

Figure. The two compositions are not the same test. Sine after inverse sine recovers x on the domain. Inverse sine after sine recovers θ only on the principal interval.

How it works

  1. Apply the forward functionShould recover x.
  2. Check y is in the principal rangeThe opposite test.
  3. If either fails, redo the pickThe check.

8Keep the claim at this chapter, not the next

Keep the claim at this chapter: principal values and partner-sums. It does not start the derivatives of inverse-trig (that sits in continuity/differentiability if taught there).

A d/dx(sin⁻¹x) dump here is a chapter-swap.

Figure. This chapter stores principal values and the standard identities. Differentiating inverse sine is a later calculus claim.

How it works

  1. Stay on range and identitiesThis chapter.
  2. Leave inverse-trig derivatives for that headingThe next map.
  3. Refuse a calculus dumpThe claim-size.
sin⁻¹(1/2) in the usual principal range is
  1. π/6
  2. 5π/6
  3. 1/sin(1/2)

Range-pick, not reciprocal.

Notes

  • The official chapter title is “Inverse Trigonometric Functions”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • y=sin⁻¹x ⇒ sin y=x, y in principal range
  • sin⁻¹x+cos⁻¹x=π/2 on [−1,1]

Recap

Hold these pegs from the official chapter “Inverse Trigonometric Functions”. The wording is ExamMaster’s teaching, not a textbook recap.

Basic Concepts
Basic concepts: y = sin⁻¹x means sin y = x with y in the taught principal range (usually [−π/2, π/2]).
Properties of Inverse Trigonometric Functions
Properties (school): sin⁻¹x + cos⁻¹x = π/2 on [−1,1] as taught; similar partner-sums for tan if listed.
A definition is a test you can run
Inverse-trig-word is a test: a principal-range pick or a partner-sum.
Name the given before the unknown
The given is a value in [−1,1] or a partner.
One worked case is enough at this class
One worked case is enough: tan⁻¹1 = π/4. Do not stack ten more arctans. The case teaches range-pick.
A miss: swapping the school name for the picture
A miss: swapping the school name for a picture.

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