Inverse Trigonometric Functions
The inverse trigonometric functions — arcsin, arccos, and arctan — reverse the process of the standard trig functions. Given a ratio, they return the angle that produces it.
This guide covers principal value ranges, domain and range restrictions, composition rules, the right-triangle method, worked examples with full solutions, and a practice quiz.
1Introduction
Standard trigonometric functions take an angle and return a ratio. For example, sin(π/6) = 1/2. But what if you know the ratio and need the angle? That’s where inverse trig functions come in.
The inverse trig functions answer the question: “What angle has this sine / cosine / tangent value?” They are essential for solving triangles, evaluating compositions, and working with trig equations.
If sin(θ) = x, then arcsin(x) = θ. The inverse “undoes” the original function — but only within a restricted range called the principal value range.
Inverse trig functions appear throughout calculus (derivatives, integrals), physics (finding angles of projection, inclines), and engineering (signal processing, robotics). Mastering them now builds a critical foundation.
2Key Definitions
Inverse Function
A function that reverses another: if f(a) = b, then f¹(b) = a.
One-to-One Function
A function where each output comes from exactly one input. Required for an inverse to exist.
Principal Value Range
The restricted range of the inverse trig function that ensures it returns exactly one angle per input.
Domain Restriction
Limiting the input of the original trig function to make it one-to-one, enabling an inverse.
arcsin x (sin¹ x)
The angle in [−π/2, π/2] whose sine equals x. Domain: [−1, 1].
arccos x (cos¹ x)
The angle in [0, π] whose cosine equals x. Domain: [−1, 1].
arctan x (tan¹ x)
The angle in (−π/2, π/2) whose tangent equals x. Domain: (−∞, ∞).
Right Triangle Method
Drawing a right triangle from an inverse trig expression to find other trig values of the resulting angle.
Interactive Inverse Trig Explorer
Drag the slider to step through key angles (0°, 30°, 45°, 60°, 90°, … 360°). Notice how the inverse functions always return principal values.
3Arcsin (Inverse Sine)
The function y = arcsin(x) answers: “What angle θ in [−π/2, π/2] has sin(θ) = x?”
| Property | Value |
|---|---|
| Domain | [−1, 1] |
| Range (Principal Values) | [−π/2, π/2] |
| Quadrants Covered | QI and QIV |
| arcsin(0) | 0 |
| arcsin(1) | π/2 |
| arcsin(−1) | −π/2 |
Common Arcsin Values
| x | arcsin(x) |
|---|---|
| −1 | −π/2 |
| −√3/2 | −π/3 |
| −√2/2 | −π/4 |
| −1/2 | −π/6 |
| 0 | 0 |
| 1/2 | π/6 |
| √2/2 | π/4 |
| √3/2 | π/3 |
| 1 | π/2 |
The restriction to [−π/2, π/2] means arcsin always returns angles in Quadrant I (for positive inputs) or Quadrant IV (for negative inputs). It never returns an angle in QII or QIII.
4Arccos (Inverse Cosine)
The function y = arccos(x) answers: “What angle θ in [0, π] has cos(θ) = x?”
| Property | Value |
|---|---|
| Domain | [−1, 1] |
| Range (Principal Values) | [0, π] |
| Quadrants Covered | QI and QII |
| arccos(1) | 0 |
| arccos(0) | π/2 |
| arccos(−1) | π |
Common Arccos Values
| x | arccos(x) |
|---|---|
| −1 | π |
| −√3/2 | 5π/6 |
| −√2/2 | 3π/4 |
| −1/2 | 2π/3 |
| 0 | π/2 |
| 1/2 | π/3 |
| √2/2 | π/4 |
| √3/2 | π/6 |
| 1 | 0 |
Complementary Relationship
arcsin(x) + arccos(x) = π/2 for all x ∈ [ −1, 1]
This identity connects arcsin and arccos. If you know one, you can find the other.
5Arctan (Inverse Tangent)
The function y = arctan(x) answers: “What angle θ in (−π/2, π/2) has tan(θ) = x?” Unlike arcsin and arccos, arctan accepts any real number as input.
| Property | Value |
|---|---|
| Domain | (−∞, ∞) |
| Range (Principal Values) | (−π/2, π/2) |
| Quadrants Covered | QI and QIV |
| arctan(0) | 0 |
| arctan(1) | π/4 |
| arctan(−1) | −π/4 |
| Horizontal Asymptotes | y = −π/2 and y = π/2 |
| lim arctan(x) as x → ∞ | π/2 |
arctan(√3) = π/3
arctan(√3/3) = π/6
arctan(−√3) = −π/3
arctan(100) ≈ 1.5608 (close to π/2)
Unlike arcsin and arccos whose domains are limited to [−1, 1], arctan accepts any real number. No matter how large or small the input, the output is always between −π/2 and π/2 (exclusive).
6Composition Rules
The most important skill with inverse trig functions is understanding what happens when you compose a trig function with its inverse (or vice versa). There are two cases, and they behave differently.
Case 1: Trig(Inverse Trig) — Always Simplifies
sin(arcsin x) = x for x ∈ [−1, 1]
cos(arccos x) = x for x ∈ [−1, 1]
tan(arctan x) = x for x ∈ (−∞, ∞)
These always simplify to x, as long as x is in the domain of the inverse function.
Case 2: Inverse Trig(Trig) — Conditional
arcsin(sin x) = x only if x ∈ [−π/2, π/2]
arccos(cos x) = x only if x ∈ [0, π]
arctan(tan x) = x only if x ∈ (−π/2, π/2)
If x is outside the principal range, you must first evaluate the inner trig function, then apply the inverse to get the principal value.
The most common exam error: assuming arcsin(sin x) = x for all x. It only works when x is in the principal range. For example, arcsin(sin(5π/6)) ≠ 5π/6. Instead: sin(5π/6) = 1/2, so arcsin(1/2) = π/6.
7Evaluating Expressions
Expressions like cos(arcsin(3/5)) or tan(arccos(4/5)) require the right triangle method. Here’s the approach:
Right Triangle Method (Step-by-Step)
- Let θ equal the inverse trig expression (e.g., θ = arcsin(3/5)).
- Interpret the ratio: sin θ = 3/5 means opposite = 3, hypotenuse = 5.
- Find the missing side using the Pythagorean theorem: adjacent = √(25 − 9) = 4.
- Read off the desired trig function: cos θ = adjacent/hypotenuse = 4/5.
Example: Find cos(arcsin(3/5))
Step 1: Let θ = arcsin(3/5), so sin θ = 3/5.
Step 2: Draw right triangle: opp = 3, hyp = 5.
Step 3: adj = √(5² − 3²) = √(25 − 9) = √16 = 4.
Step 4: cos θ = adj/hyp = 4/5.
Example: Find sin(arctan(7/24))
Step 1: Let θ = arctan(7/24), so tan θ = 7/24.
Step 2: Draw right triangle: opp = 7, adj = 24.
Step 3: hyp = √(7² + 24²) = √(49 + 576) = √625 = 25.
Step 4: sin θ = opp/hyp = 7/25.
When the input to the inverse trig function is negative, the angle θ may be in QIV. The sign of the adjacent side (or the appropriate side) matters. Always consider which quadrant the principal value falls in.
8Key Formulas
Domain & Range Summary
| Function | Domain | Range |
|---|---|---|
| arcsin x | [−1, 1] | [−π/2, π/2] |
| arccos x | [−1, 1] | [0, π] |
| arctan x | (−∞, ∞) | (−π/2, π/2) |
Complementary Identity
arcsin(x) + arccos(x) = π/2
Negative Argument Identities
arcsin(−x) = −arcsin(x)
arccos(−x) = π − arccos(x)
arctan(−x) = −arctan(x)
Reciprocal Argument (for arctan)
arctan(x) + arctan(1/x) = π/2 (for x > 0)
9Worked Examples
Example 1: Evaluate arccos(cos(5π/3))
Step 1: Is 5π/3 in the principal range [0, π]? No (5π/3 ≈ 5.24 > π ≈ 3.14).
Step 2: Evaluate the inner function: cos(5π/3) = cos(2π − π/3) = cos(π/3) = 1/2.
Step 3: Now evaluate: arccos(1/2) = π/3.
Example 2: Evaluate tan(arcsin(−5/13))
Step 1: Let θ = arcsin(−5/13). Then sin θ = −5/13, and θ is in QIV.
Step 2: opp = −5, hyp = 13. adj = √(169 − 25) = √144 = 12 (positive in QIV).
Step 3: tan θ = opp/adj = −5/12 = −5/12.
Example 3: Simplify sin(arccos(x)) in terms of x
Step 1: Let θ = arccos(x). Then cos θ = x = x/1. Think: adj = x, hyp = 1.
Step 2: opp = √(1 − x²) by Pythagorean theorem.
Step 3: sin θ = opp/hyp = √(1 − x²).
Note: The result is always non-negative because arccos returns angles in [0, π] where sine ≥ 0.
Example 4: Evaluate arcsin(sin(7π/6))
Step 1: Is 7π/6 in [−π/2, π/2]? No.
Step 2: sin(7π/6) = −1/2 (reference angle π/6, QIII).
Step 3: arcsin(−1/2) = −π/6 (the angle in [−π/2, π/2] with sine = −1/2).
Example 5: Find cos(2 arctan(3/4))
Step 1: Let θ = arctan(3/4). Build triangle: opp = 3, adj = 4, hyp = 5.
Step 2: cos θ = 4/5, sin θ = 3/5.
Step 3: Use double-angle: cos(2θ) = cos²θ − sin²θ = 16/25 − 9/25 = 7/25.
10Memory Aids
Arcsin and arcTangent both use the range [−π/2, π/2], covering Quadrants I and IV. ArcCosine is the odd one out with [0, π] (QI and QII).
sin(arcsin x) = x always works (when x is in the domain). But arcsin(sin x) = x only when x is already in the principal range.
When you see trig(inverse trig), draw a right triangle. Label the sides from the inner inverse trig, find the missing side with the Pythagorean theorem, then read off the outer trig function.
The notation sin¹(x) means arcsin(x), NOT (sin x)¹ = csc x. This is a notational trap — the −1 is not an exponent, it indicates the inverse function.
11Common Mistakes to Avoid
Assuming arcsin(sin x) = x for All x
Wrong: arcsin(sin(2π/3)) = 2π/3
Right: sin(2π/3) = √3/2. arcsin(√3/2) = π/3. The input 2π/3 is outside [−π/2, π/2].
Confusing sin¹(x) with 1/sin(x)
Wrong: sin¹(0.5) = 1/sin(0.5) = 2.086...
Right: sin¹(0.5) = arcsin(0.5) = π/6. The reciprocal of sin is csc, not arcsin.
Forgetting Domain Restrictions
Wrong: arcsin(2) = some angle
Right: arcsin(2) is undefined. The domain of arcsin is [−1, 1], and 2 is outside it.
Wrong Quadrant for arccos
Wrong: arccos(−1/2) = −π/3
Right: arccos always returns values in [0, π]. arccos(−1/2) = 2π/3 (QII, not QIV).
Ignoring Sign in Right Triangle Method
Wrong: For arctan(−3/4), drawing all sides positive.
Right: Since the angle is in QIV (arctan of a negative), the opposite side is negative. This affects the signs of sine and cosecant.
Mixing Up Range of arcsin vs arccos
Wrong: arcsin returns values in [0, π].
Right: arcsin: [−π/2, π/2]. arccos: [0, π]. They are complementary, not interchangeable.
12Quick Revision Summary
- arcsin x: domain [−1, 1], range [−π/2, π/2] (QI & QIV).
- arccos x: domain [−1, 1], range [0, π] (QI & QII).
- arctan x: domain (−∞, ∞), range (−π/2, π/2) (QI & QIV).
- trig(inverse trig) always simplifies: sin(arcsin x) = x.
- inverse trig(trig) only simplifies when the input is in the principal range.
- Use the right triangle method for expressions like cos(arcsin x).
- arcsin(x) + arccos(x) = π/2 for all x in [−1, 1].
- arcsin and arctan are odd functions: f(−x) = −f(x).
- sin¹ x means arcsin, not 1/sin x (that’s csc x).
- Inputs outside the domain are undefined: arcsin(1.5) does not exist.
Frequently Asked Questions
Why restrict the domain?
Trig functions are periodic and not one-to-one. We restrict to make them one-to-one so the inverse is a proper function.
What's the difference between arcsin x and sin⁻¹ x?
Same thing! Different notation. Be careful: sin⁻¹ x ≠ 1/sin x (that's csc x).
How do I remember the principal value ranges?
arcsin and arctan: [−π/2, π/2] (centered on y-axis, QI and QIV). arccos: [0, π] (top half, QI and QII).
What if arcsin(sin x) ≠ x?
This happens when x is outside [−π/2, π/2]. Find sin x first, then find the angle in [−π/2, π/2] with that same sine.
Practice Quiz
Test your knowledge — select the correct answer for each question.
1.What is the domain of y = arcsin x?
2.What is the range of y = arccos x?
3.Evaluate arcsin(√3/2)
4.Evaluate arctan(−1)
5.Evaluate cos(arccos(0.7))
6.Evaluate tan(arctan(−10))
7.Evaluate arcsin(sin(π/2))
8.Evaluate arccos(cos(4π/3))
9.Which expression is undefined?
10.If θ = arctan(5/12), what is cos θ?
Study Tips
- Memorize the three domain/range pairs — this is the foundation for everything else with inverse trig.
- Practice the right triangle method — draw the triangle every time until it becomes automatic.
- Check both composition directions — always ask “Is the inner value in the principal range?” before simplifying.
- Use the complementary identity — arcsin(x) + arccos(x) = π/2 can save time on exams.
- Watch the notation — train yourself to read sin¹ as “arcsin” and never as “1 over sine.”