Class 12 Mathematics INVERSE CIRCULAR FUNCTIONS Notes

Class 12 Mathematics INVERSE CIRCULAR FUNCTIONS Notes
Class 12 Mathematics Inverse Circular Functions Notes | Nepal eNotes

Unit 8

Trigonometry

Class 12 Mathematics

Inverse Circular Functions

Class 12 Mathematics – Inverse Circular Functions Notes PDF

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Contents / Quick Navigation

1. Introduction 2. Principal Values 3. Domains and Ranges 4. Fundamental Relations 5. Composition Relations 6. Relations of tan⁻¹ 7. Useful Transformations 8. Worked Examples 9. Common Mistakes 10. Important Exam Questions 11. One-Minute Revision 12. Diagram Practice

NEB / CDC Focus

Inverse circular functions are inverse forms of trigonometric functions after their domains are suitably restricted so that they become one-to-one. The principal focus is to define these inverse functions and establish standard relations among them.

1. Introduction

Inverse Circular Function

If a trigonometric function is restricted to a domain on which it is one-to-one, its inverse is called an inverse circular or inverse trigonometric function.

For example, sin x is not one-to-one on all real numbers. But on the interval [-π/2,π/2] it is one-to-one and maps onto [-1,1]. Hence its inverse is defined by

y=sin⁻¹x   ⇔   x=sin y,   −1≤x≤1,   −π/2≤y≤π/2.
Very Important

sin⁻¹x means the inverse sine function, not 1/sin x. The reciprocal of sine is cosec x.

Why sin x Needs a Restricted Domain −π/2 π/2 On [−π/2,π/2], sine is one-to-one and therefore invertible.
Figure 1: The highlighted principal branch is used to define sin⁻¹x.

2. Principal Values

A trigonometric equation such as sin θ=1/2 has infinitely many solutions. An inverse trigonometric function must return exactly one value, called the principal value.

Principal Value

The principal value is the unique angle selected from the prescribed principal range of the inverse circular function.

Examples:

  • sin⁻¹(1/2)=π/6, not 5π/6.
  • cos⁻¹(−1/2)=2π/3.
  • tan⁻¹(1)=π/4.
Principal-Angle Ranges on the Unit Circle sin⁻¹ range: −π/2 to π/2 cos⁻¹ range: 0 to π tan⁻¹ range: −π/2 to π/2 Endpoints are treated according to each function’s standard principal range.
Figure 2: Principal ranges ensure a unique inverse value.

3. Domains and Principal Ranges

Inverse functionDomainPrincipal rangeMeaning
sin⁻¹x [-1,1] [-π/2,π/2] sin(sin⁻¹x)=x
cos⁻¹x [-1,1] [0,π] cos(cos⁻¹x)=x
tan⁻¹x ℝ (−π/2,π/2) tan(tan⁻¹x)=x
cot⁻¹x ℝ (0,π) cot(cot⁻¹x)=x
sec⁻¹x (−∞,−1]∪[1,∞) [0,π], θ≠π/2 sec(sec⁻¹x)=x
cosec⁻¹x (−∞,−1]∪[1,∞) [-π/2,π/2], θ≠0 cosec(cosec⁻¹x)=x
Convention Note

Textbooks can differ in the chosen principal interval for cot⁻¹, sec⁻¹ and cosec⁻¹. The table above uses the common school convention; for an exam, follow the convention used by your prescribed text/teacher.

3.1 Graph of y=sin⁻¹x

Graph of y = sin⁻¹x −1 1 π/2 −π/2 Domain [-1,1], range [-π/2,π/2].
Figure 3: The graph of sin⁻¹x is the reflection of the principal sine branch in y=x.

3.2 Graph of y=cos⁻¹x

Graph of y = cos⁻¹x −1 1 π Domain [-1,1], range [0,π]. The function is decreasing.
Figure 4: cos⁻¹x decreases from π to 0 as x goes from −1 to 1.

3.3 Graph of y=tan⁻¹x

Graph of y = tan⁻¹x π/2 −π/2 Domain ℝ; horizontal asymptotes y=±π/2.
Figure 5: tan⁻¹x is increasing and its principal values stay strictly between −π/2 and π/2.

4. Fundamental Relations Between Inverse Circular Functions

4.1 Complementary Relations

Core Relations
sin⁻¹x + cos⁻¹x = π/2,   −1≤x≤1 tan⁻¹x + cot⁻¹x = π/2

Proof of sin⁻¹x+cos⁻¹x=π/2

Let

sin⁻¹x=θ.

Then sinθ=x, with −π/2≤θ≤π/2.

Using cos(π/2−θ)=sinθ=x,

cos⁻¹x=π/2−θ.

Therefore

sin⁻¹x+cos⁻¹x=π/2.

Complementary-Angle Idea θ π/2−θ complementary acute angles opposite hypotenuse sinθ = cos(π/2−θ)
Figure 6: Sine and cosine of complementary angles lead to the inverse relation.

4.2 Sign Relations

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

The first two reflect odd symmetry; the cosine relation follows from its principal range [0,π].

4.3 Reciprocal-Type Relations

sec⁻¹x = cos⁻¹(1/x),   |x|≥1 cosec⁻¹x = sin⁻¹(1/x),   |x|≥1

5. Composition Relations

Direct Inverse Compositions
sin(sin⁻¹x)=x,   −1≤x≤1 cos(cos⁻¹x)=x,   −1≤x≤1 tan(tan⁻¹x)=x,   x∈ℝ

The reverse-looking compositions require care because the original trigonometric functions are periodic:

Principal-Range Warning

sin⁻¹(sinθ)=θ is true only when θ∈[-π/2,π/2]. Outside this interval, the result is the principal angle having the same sine.

ExpressionWhen it simplifies directly
sin⁻¹(sinθ)θ∈[-π/2,π/2]
cos⁻¹(cosθ)θ∈[0,π]
tan⁻¹(tanθ)θ∈(−π/2,π/2)
Composition: Function and Inverse Principal angle θ −π/2≤θ≤π/2 sin x=sinθ sin⁻¹ θ The return to θ works because θ begins inside the principal range.
Figure 7: An inverse undoes the original function when the angle is inside the chosen principal branch.

6. Important Relations Involving tan⁻¹

6.1 Addition Formula

Let

A=tan⁻¹x,   B=tan⁻¹y.

Then tan A=x, tan B=y. Hence

tan(A+B)= x+y1−xy.

Therefore, with the correct principal-value adjustment,

tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy))

when the sum lies in the principal range; if not, add or subtract π as required by the signs of x and y and the quadrant of A+B.

6.2 Difference Formula

tan⁻¹x − tan⁻¹y = tan⁻¹((x−y)/(1+xy))

again with principal-value care.

6.3 Double-Angle Form

2tan⁻¹x = tan⁻¹(2x/(1−x²))

subject to the appropriate principal-value adjustment.

Worked Example

Evaluate tan⁻¹(1/2)+tan⁻¹(1/3).

tan(A+B)= [(1/2)+(1/3)]/[1−(1/2)(1/3)] =(5/6)/(5/6)=1.

Both angles are positive acute principal values, so A+B=π/4.

Answer: π/4.

tan⁻¹ Addition: Do Not Ignore the Quadrant Compute (x+y)/(1−xy) candidate tangent Check signs and sizes What quadrant is A+B in? A=tan⁻¹x, B=tan⁻¹y Choose principal value A tangent value alone does not uniquely determine an angle.
Figure 8: Principal-value adjustment is essential in inverse-tangent identities.

7. Useful Transformations

7.1 Converting sin⁻¹ to tan⁻¹

Let θ=sin⁻¹x. Then sinθ=x and, on the principal interval, cosθ=√(1−x²). Thus

sin⁻¹x = tan⁻¹(x/√(1−x²)),   |x|<1.

7.2 Converting cos⁻¹ to tan⁻¹

For a principal acute case, if θ=cos⁻¹x, then tanθ=√(1−x²)/x. The quadrant must be handled carefully when x<0.

7.3 Half-Angle Type Relation

tan⁻¹x = sin⁻¹(x/√(1+x²)),

for real x, with the principal ranges stated above.

7.4 Standard Evaluation Strategy

  1. Replace an inverse expression by an angle, e.g. θ=sin⁻¹x.
  2. Translate it into an ordinary trigonometric equation, e.g. sinθ=x.
  3. Use a right triangle or identity to find the required sine, cosine or tangent.
  4. Check the principal range before fixing the sign.
  5. Convert back to the required inverse function if necessary.

8. Worked Examples

Example 1: Principal value

Evaluate sin⁻¹(−√3/2).

The sine is −√3/2 at principal angle −π/3.

Answer: −π/3.

Example 2: Evaluate cos⁻¹(cos 5π/3)

cos(5π/3)=1/2.

Since the principal range of cos⁻¹ is [0,π],

cos⁻¹(1/2)=π/3.

Answer: π/3, not 5π/3.

Example 3: Evaluate sin(sin⁻¹(3/5))

Direct inverse composition gives 3/5.

Example 4: Evaluate cos(sin⁻¹(3/5))

Let θ=sin⁻¹(3/5). Then sinθ=3/5.

Since θ∈[-π/2,π/2], cosθ≥0.

cosθ=√(1−9/25)=4/5.

Answer: 4/5.

Example 5: Prove a complementary identity

Show that sin⁻¹(3/5)+cos⁻¹(3/5)=π/2.

This follows directly from sin⁻¹x+cos⁻¹x=π/2 for −1≤x≤1.

Example 6: tan⁻¹ addition

Evaluate tan⁻¹1+tan⁻¹2.

Let A=tan⁻¹1=π/4 and B=tan⁻¹2. Both are positive, and A+B>π/2 because B>π/4.

tan(A+B)=(1+2)/(1−2)=−3.

Since A+B lies in quadrant II,

A+B=π−tan⁻¹3.

Example 7: A transformed inverse expression

Evaluate tan(sin⁻¹x), for |x|<1.

Let θ=sin⁻¹x. Then sinθ=x and cosθ=√(1−x²).

tanθ=x/√(1−x²).

Therefore tan(sin⁻¹x)=x/√(1−x²).

Example 8: Solve an inverse equation

Solve sin⁻¹x=π/6.

Apply sine to both sides:

x=sin(π/6)=1/2.

Answer: x=1/2.

Example 9: Domain check

Is sin⁻¹(3/2) real?

No. The real domain of sin⁻¹x is [-1,1], and 3/2 lies outside it.

Example 10: Use odd symmetry

Simplify tan⁻¹(−x)+tan⁻¹x.

Since tan⁻¹(−x)=−tan⁻¹x,

tan⁻¹(−x)+tan⁻¹x=0.

9. Quick Relation Table

RelationCondition / note
sin⁻¹x+cos⁻¹x=π/2−1≤x≤1
sin⁻¹(−x)=−sin⁻¹x−1≤x≤1
cos⁻¹(−x)=π−cos⁻¹x−1≤x≤1
tan⁻¹(−x)=−tan⁻¹xx∈ℝ
tan⁻¹x+cot⁻¹x=π/2under the stated principal-range convention
sec⁻¹x=cos⁻¹(1/x)|x|≥1
cosec⁻¹x=sin⁻¹(1/x)|x|≥1
sin(sin⁻¹x)=xx∈[-1,1]
cos(cos⁻¹x)=xx∈[-1,1]
tan(tan⁻¹x)=xx∈ℝ

10. Common Mistakes and Warnings

Mistake 1: Reciprocal confusion

sin⁻¹x is not cosec x.

Mistake 2: Ignoring principal range

An inverse circular function returns one principal angle, not every angle with the same trig value.

Mistake 3: sin⁻¹(sinθ)=θ always

This direct simplification is valid only when θ lies in the principal range of sin⁻¹.

Mistake 4: Domain of sin⁻¹ and cos⁻¹

Real-valued inputs must satisfy −1≤x≤1.

Mistake 5: tan⁻¹ addition without quadrant check

The tangent formula may require a ±π correction.

Mistake 6: Wrong sign for cos(sin⁻¹x)

On the principal sine range, cosine is non-negative, so use +√(1−x²).

Mistake 7: Degrees/radians mixing

Use radians in standard formulae unless the problem explicitly uses degrees.

Mistake 8: Forgetting conventions

Principal ranges for cot⁻¹, sec⁻¹ and cosec⁻¹ may vary by textbook; use the prescribed convention consistently.

11. Exam-Important Formula Sheet

Core Formulae
sin⁻¹x+cos⁻¹x=π/2 tan⁻¹x+cot⁻¹x=π/2 sin⁻¹(−x)=−sin⁻¹x cos⁻¹(−x)=π−cos⁻¹x tan⁻¹(−x)=−tan⁻¹x tan⁻¹x±tan⁻¹y → use tan(A±B) and check the principal value sec⁻¹x=cos⁻¹(1/x), |x|≥1 cosec⁻¹x=sin⁻¹(1/x), |x|≥1 sin(sin⁻¹x)=x cos(cos⁻¹x)=x tan(tan⁻¹x)=x tan(sin⁻¹x)=x/√(1−x²), |x|<1

12. Important Exam Questions

Short-Answer Questions

  1. Define an inverse circular function.
  2. Why must the domain of a trigonometric function be restricted before defining its inverse?
  3. State the domain and principal range of sin⁻¹x.
  4. State the domain and principal range of cos⁻¹x.
  5. State the domain and principal range of tan⁻¹x.
  6. Evaluate standard principal values such as sin⁻¹(1/2), cos⁻¹(−1/2) and tan⁻¹1.
  7. State the relation between sin⁻¹x and cos⁻¹x.
  8. State the odd/even-type sign relations for inverse sine, cosine and tangent.

Proof / Long-Answer Questions

  1. Prove that sin⁻¹x+cos⁻¹x=π/2.
  2. Establish sin⁻¹(−x)=−sin⁻¹x.
  3. Establish cos⁻¹(−x)=π−cos⁻¹x.
  4. Derive the addition relation for tan⁻¹x+tan⁻¹y and discuss the principal-value condition.
  5. Derive a relation between sin⁻¹x and tan⁻¹.

Numerical / Simplification Questions

  1. Evaluate expressions such as sin⁻¹(sinθ), taking the principal range into account.
  2. Evaluate cos(sin⁻¹x) or tan(cos⁻¹x).
  3. Simplify sums and differences of inverse tangent expressions.
  4. Solve simple equations involving one inverse circular function.
  5. Test whether a given real number lies in the domain of an inverse circular function.

Diagram / Graph Questions

  1. Sketch the restricted sine branch used to define sin⁻¹x.
  2. Sketch y=sin⁻¹x.
  3. Sketch y=cos⁻¹x.
  4. Sketch y=tan⁻¹x and indicate its horizontal asymptotes.
  5. Show principal-angle ranges on a unit-circle diagram.

13. One-Minute Revision

Quick Revision
  • Inverse circular functions are defined after restricting trigonometric functions to one-to-one branches.
  • sin⁻¹x is not the reciprocal of sine.
  • Domain of sin⁻¹x and cos⁻¹x is [-1,1].
  • Principal range of sin⁻¹x is [-π/2,π/2].
  • Principal range of cos⁻¹x is [0,π].
  • Principal range of tan⁻¹x is (−π/2,π/2).
  • sin⁻¹x+cos⁻¹x=π/2.
  • sin⁻¹(−x)=−sin⁻¹x.
  • cos⁻¹(−x)=π−cos⁻¹x.
  • tan⁻¹(−x)=−tan⁻¹x.
  • sin(sin⁻¹x)=x and similarly for cosine and tangent on their inverse domains.
  • The reverse composition simplifies directly only when the angle lies in the inverse function’s principal range.
  • Inverse tangent addition formulae require a quadrant/principal-value check.
  • For θ=sin⁻¹x, cosθ=√(1−x²) on the principal sine interval.
  • Always state or respect the domain and principal range when establishing inverse-function relations.

14. Diagram Practice

  • Restricted sine graph on [-π/2,π/2].
  • Principal ranges on the unit circle.
  • Graph of y=sin⁻¹x.
  • Graph of y=cos⁻¹x.
  • Graph of y=tan⁻¹x with asymptotes.
  • Right-triangle diagram for complementary inverse relations.
  • Function/inverse composition diagram.
  • Principal-value decision diagram for inverse-tangent addition.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
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