Inverse Trigonometric Functions: Definitions, Formulas, Graphs, and Applications

Inverse trigonometric functions solve a fundamental question in mathematics: given the value of a trigonometric function, what is the corresponding angle? For example, we know that sin 30° = 0.5, but what if we know that the sine value is 0.5 and want to find the angle? This is precisely where inverse trigonometric functions come into play.

This article provides a systematic overview of inverse trigonometric functions. If you need to quickly compute inverse trigonometric values, try our Inverse Trigonometric Calculator, which supports precise calculations for arcsin, arccos, arctan, and more.

1. Why Do We Need Inverse Trigonometric Functions?

1.1 The Motivation

In practical applications, we frequently encounter problems like:

• Given a rope’s length and the height of its attachment point, find the angle between the rope and the ground
• Given a building’s height and the observation distance, find the angle of elevation
• Given the rise and horizontal run of a slope, find the slope angle

The common thread in all these problems is: given a ratio, find the angle. Trigonometric functions provide the mapping from “angle → ratio,” while inverse trigonometric functions provide the reverse mapping from “ratio → angle.”

1.2 Conditions for an Inverse Function

To construct an inverse function, the original function must be one-to-one (injective). However, trigonometric functions are periodic and naturally not one-to-one — for example, sin 0° = sin 180° = 0, meaning a single function value corresponds to infinitely many angles.

To resolve this, mathematicians restrict the domain of trigonometric functions so that they become one-to-one on a specific interval, thereby allowing an inverse function to be defined. This restricted interval is called the principal value range.

2. The Six Inverse Trigonometric Functions

2.1 Arcsine (arcsin)

$y = \arcsin x \quad \Leftrightarrow \quad x = \sin y, \quad -1 \leq x \leq 1, \quad -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

• Domain: [-1, 1]
• Range (Principal Value): [-π/2, π/2], i.e., [-90°, 90°]
• Meaning: Given x, returns an angle whose sine equals x and lies within [-π/2, π/2]

2.2 Arccosine (arccos)

$y = \arccos x \quad \Leftrightarrow \quad x = \cos y, \quad -1 \leq x \leq 1, \quad 0 \leq y \leq \pi$

• Domain: [-1, 1]
• Range (Principal Value): [0, π], i.e., [0°, 180°]
• Meaning: Given x, returns an angle whose cosine equals x and lies within [0, π]

2.3 Arctangent (arctan)

$y = \arctan x \quad \Leftrightarrow \quad x = \tan y, \quad x \in \mathbb{R}, \quad -\frac{\pi}{2} < y < \frac{\pi}{2}$

• Domain: All real numbers (-∞, +∞)
• Range (Principal Value): (-π/2, π/2), i.e., (-90°, 90°)
• Meaning: Given x, returns an angle whose tangent equals x and lies within (-π/2, π/2)

2.4 Arccotangent (arccot)

$y = \text{arccot}\, x \quad \Leftrightarrow \quad x = \cot y, \quad x \in \mathbb{R}, \quad 0 < y < \pi$

• Domain: All real numbers (-∞, +∞)
• Range (Principal Value): (0, π), i.e., (0°, 180°)
• Meaning: Given x, returns an angle whose cotangent equals x and lies within (0, π)

2.5 Arcsecant (arcsec)

$y = \text{arcsec}\, x \quad \Leftrightarrow \quad x = \sec y, \quad |x| \geq 1, \quad y \in [0, \pi], \, y \neq \frac{\pi}{2}$

• Domain: (-∞, -1] ∪ [1, +∞)
• Range (Principal Value): [0, π], excluding y = π/2
• Meaning: Given x, returns an angle whose secant equals x and lies within [0, π] (excluding π/2)

2.6 Arccosecant (arccsc)

$y = \text{arccsc}\, x \quad \Leftrightarrow \quad x = \csc y, \quad |x| \geq 1, \quad y \in [-\frac{\pi}{2}, \frac{\pi}{2}], \, y \neq 0$

• Domain: (-∞, -1] ∪ [1, +∞)
• Range (Principal Value): [-π/2, π/2], excluding y = 0
• Meaning: Given x, returns an angle whose cosecant equals x and lies within [-π/2, π/2] (excluding 0)

2.7 Domain and Range Summary

Function	Domain	Range (Principal Value)
arcsin x	[-1, 1]	[-π/2, π/2]
arccos x	[-1, 1]	[0, π]
arctan x	(-∞, +∞)	(-π/2, π/2)
arccot x	(-∞, +∞)	(0, π)
arcsec x	(-∞, -1] ∪ [1, +∞)	[0, π], y ≠ π/2
arccsc x	(-∞, -1] ∪ [1, +∞)	[-π/2, π/2], y ≠ 0

Use our Inverse Trigonometric Calculator to quickly compute any inverse trigonometric function value and get results in both degrees and radians.

3. Special Values

The following are commonly used special values of inverse trigonometric functions that are worth memorizing:

3.1 Special Values of arcsin

x	arcsin x (radians)	arcsin x (degrees)
-1	-π/2	-90°
-√3/2	-π/3	-60°
-√2/2	-π/4	-45°
-1/2	-π/6	-30°
0	0	0°
1/2	π/6	30°
√2/2	π/4	45°
√3/2	π/3	60°
1	π/2	90°

3.2 Special Values of arccos

x	arccos x (radians)	arccos x (degrees)
-1	π	180°
-√3/2	5π/6	150°
-√2/2	3π/4	135°
-1/2	2π/3	120°
0	π/2	90°
1/2	π/3	60°
√2/2	π/4	45°
√3/2	π/6	30°
1	0	0°

3.3 Special Values of arctan

x	arctan x (radians)	arctan x (degrees)
-√3	-π/3	-60°
-1	-π/4	-45°
-√3/3	-π/6	-30°
0	0	0°
√3/3	π/6	30°
1	π/4	45°
√3	π/3	60°

💡 Memory Tip: The special values of arcsin and arccos are complementary, meaning arcsin x + arccos x = π/2. For example, arcsin(1/2) = π/6 and arccos(1/2) = π/3, and their sum is exactly π/2.

4. Properties of Inverse Trigonometric Functions

4.1 Monotonicity

Function	Monotonicity
arcsin x	Strictly increasing on [-1, 1]
arccos x	Strictly decreasing on [-1, 1]
arctan x	Strictly increasing on (-∞, +∞)
arccot x	Strictly decreasing on (-∞, +∞)
arcsec x	Strictly increasing on each interval of its domain
arccsc x	Strictly decreasing on each interval of its domain

4.2 Symmetry (Odd/Even)

• Odd functions: arcsin(-x) = -arcsin x, arctan(-x) = -arctan x, arccsc(-x) = -arccsc x
• Neither odd nor even: arccos x, arccot x, arcsec x (because their ranges are not symmetric about the origin)

4.3 Asymptotic Behavior

• As x → +∞, arctan x → π/2; as x → -∞, arctan x → -π/2
• As x → +∞, arccot x → 0; as x → -∞, arccot x → π

Thus, y = π/2 and y = -π/2 are horizontal asymptotes of arctan, while y = 0 and y = π are horizontal asymptotes of arccot.

5. Graphs of Inverse Trigonometric Functions

5.1 y = arcsin x

• Shape: An S-shaped curve rising from lower left to upper right
• Endpoints: (-1, -π/2) and (1, π/2)
• Passes through origin: arcsin 0 = 0
• Symmetry: Symmetric about the origin (odd function)
• Relationship to y = sin x: The graph of y = arcsin x is the reflection of y = sin x (restricted to [-π/2, π/2]) about the line y = x

5.2 y = arccos x

• Shape: A decreasing curve from upper left to lower right
• Endpoints: (-1, π) and (1, 0)
• Notable point: arccos 0 = π/2
• Relationship to y = cos x: The graph of y = arccos x is the reflection of y = cos x (restricted to [0, π]) about the line y = x

5.3 y = arctan x

• Shape: A stretched S-shaped curve, strictly increasing from left to right
• Passes through origin: arctan 0 = 0
• Horizontal asymptotes: y = π/2 (above) and y = -π/2 (below)
• Symmetry: Symmetric about the origin (odd function)
• Key feature: Domain is all real numbers, but range is confined to the open interval (-π/2, π/2)

5.4 y = arccot x

• Shape: A strictly decreasing curve from left to right
• Notable point: arccot 0 = π/2
• Horizontal asymptotes: y = 0 (to the right) and y = π (to the left)
• Key feature: Domain is all real numbers, range is the open interval (0, π)

6. Core Formulas and Identities

6.1 Complementary Relationships

$\arcsin x + \arccos x = \frac{\pi}{2}, \quad -1 \leq x \leq 1$

$\arctan x + \text{arccot}\, x = \frac{\pi}{2}, \quad x \in \mathbb{R}$

$\text{arcsec}\, x + \text{arccsc}\, x = \frac{\pi}{2}, \quad |x| \geq 1$

💡 These three complementary relationships tell us that each pair of “co-” inverse trigonometric functions always sums to π/2 (i.e., 90°), mirroring the complementary relationship between sine and cosine.

6.2 Negative Argument Identities

$\arcsin(-x) = -\arcsin x, \quad -1 \leq x \leq 1$

$\arccos(-x) = \pi - \arccos x, \quad -1 \leq x \leq 1$

$\arctan(-x) = -\arctan x, \quad x \in \mathbb{R}$

$\text{arccot}(-x) = \pi - \text{arccot}\, x, \quad x \in \mathbb{R}$

6.3 Reciprocal Relationships

$\arcsin x = \arctan \frac{x}{\sqrt{1-x^2}}, \quad -1 < x < 1$

$\arccos x = \arctan \frac{\sqrt{1-x^2}}{x}, \quad 0 < x \leq 1$

$\text{arcsec}\, x = \arccos \frac{1}{x}, \quad |x| \geq 1$

$\text{arccsc}\, x = \arcsin \frac{1}{x}, \quad |x| \geq 1$

6.4 Composition Rules

When trigonometric functions are composed with their inverses:

$\sin(\arcsin x) = x, \quad -1 \leq x \leq 1$

$\cos(\arccos x) = x, \quad -1 \leq x \leq 1$

$\tan(\arctan x) = x, \quad x \in \mathbb{R}$

However, the reverse compositions have important restrictions:

$\arcsin(\sin x) = x, \quad \text{only when } -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

$\arccos(\cos x) = x, \quad \text{only when } 0 \leq x \leq \pi$

$\arctan(\tan x) = x, \quad \text{only when } -\frac{\pi}{2} < x < \frac{\pi}{2}$

⚠️ Common Pitfall: arcsin(sin 5π/6) ≠ 5π/6, because 5π/6 is not within arcsin’s range [-π/2, π/2]. The correct answer is arcsin(sin 5π/6) = arcsin(1/2) = π/6.

6.5 Addition Formula for Arctangent

$\arctan a + \arctan b = \arctan \frac{a+b}{1-ab} + \begin{cases} 0, & ab < 1 \\ \pi, & ab > 1, \, a > 0 \\ -\pi, & ab > 1, \, a < 0 \end{cases}$

This formula has important applications in computing series expansions for π, such as Machin’s famous formula:

$\frac{\pi}{4} = 4\arctan\frac{1}{5} - \arctan\frac{1}{239}$

7. Derivatives and Integrals

7.1 Derivative Formulas

The derivatives of inverse trigonometric functions are extremely important in calculus:

$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}, \quad -1 < x < 1$

$\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}, \quad -1 < x < 1$

$\frac{d}{dx}\arctan x = \frac{1}{1+x^2}, \quad x \in \mathbb{R}$

$\frac{d}{dx}\text{arccot}\, x = -\frac{1}{1+x^2}, \quad x \in \mathbb{R}$

$\frac{d}{dx}\text{arcsec}\, x = \frac{1}{|x|\sqrt{x^2-1}}, \quad |x| > 1$

$\frac{d}{dx}\text{arccsc}\, x = -\frac{1}{|x|\sqrt{x^2-1}}, \quad |x| > 1$

💡 Note: The derivatives of arcsin and arccos are negatives of each other; arctan and arccot are negatives of each other; arcsec and arccsc are negatives of each other. This is consistent with their complementary relationships.

7.2 Integration Formulas

The following are common integrals involving inverse trigonometric functions:

$\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C$

$\int \frac{1}{1+x^2} \, dx = \arctan x + C$

$\int \frac{1}{x\sqrt{x^2-1}} \, dx = \text{arcsec}\, |x| + C$

More generally:

$\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin \frac{x}{a} + C, \quad a > 0$

$\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\arctan \frac{x}{a} + C, \quad a > 0$

8. Trigonometric Values of Inverse Trigonometric Expressions

In certain calculations, we need to find the trigonometric function values of arcsin, arccos, or arctan results. Here are several useful identities:

Given θ = arcsin x, find other trig values:

$\sin(\arcsin x) = x$

$\cos(\arcsin x) = \sqrt{1 - x^2}$

$\tan(\arcsin x) = \frac{x}{\sqrt{1 - x^2}}$

Given θ = arccos x, find other trig values:

$\cos(\arccos x) = x$

$\sin(\arccos x) = \sqrt{1 - x^2}$

$\tan(\arccos x) = \frac{\sqrt{1 - x^2}}{x}$

Given θ = arctan x, find other trig values:

$\tan(\arctan x) = x$

$\sin(\arctan x) = \frac{x}{\sqrt{1 + x^2}}$

$\cos(\arctan x) = \frac{1}{\sqrt{1 + x^2}}$

9. Real-World Applications

Inverse trigonometric functions have extensive applications in science, engineering, and daily life.

9.1 Physics

• Inclined plane problems: Given the height h and length l of a ramp, the inclination angle is θ = arcsin(h/l)
• Optics: In Snell’s law of refraction n₁sin θ₁ = n₂sin θ₂, finding the refraction angle requires the arcsine function
• Mechanics: When analyzing the direction of a resultant force, arctan is commonly used to find the angle with a coordinate axis

9.2 Engineering

• Surveying: Surveyors calculate slope angles by measuring horizontal distances and elevation differences
• Electrical Engineering: In AC circuits, the impedance angle is calculated as φ = arctan(X_L / R)
• Mechanical Design: Pressure angle calculations in gear transmissions

9.3 Computer Science

• Computer Graphics: The atan2(y, x) function is used to compute the azimuth angle of a point in the 2D plane, widely used in rotation transformations and direction calculations
• Robotics: Inverse kinematics uses inverse trigonometric functions to calculate joint angles
• Navigation Systems: Computing bearing angles and distances from latitude/longitude differences

9.4 Everyday Life

• Building Measurement: Calculating building height using the angle of elevation and observation distance, θ = arctan(h/d)
• Slope Calculation: Road gradient angle = arctan(rise / horizontal run)
• Photography: Calculating a camera’s field of view requires inverse trigonometric functions

10. Common Mistakes and Misconceptions

10.1 arcsin vs. sin⁻¹

In mathematics, arcsin x and sin⁻¹ x refer to the same concept — the inverse sine function. However, it’s crucial to note:

$\sin^{-1} x \neq \frac{1}{\sin x}$

$\frac{1}{\sin x}$ should be written as $(\sin x)^{-1}$ or $\csc x$ (the cosecant function).

10.2 The Importance of Range Restrictions

Because inverse trigonometric functions are restricted to their principal value ranges, the result directly obtained from a calculator may not be the answer required by a given problem.

Example: Given that sin θ = 0.5 and θ is in the second quadrant, find θ.

• arcsin(0.5) = π/6 = 30° (the direct calculator result)
• Since θ must be in the second quadrant, the actual answer is θ = π - π/6 = 5π/6 = 150°

10.3 The atan2 Function

In programming, the standard arctan function can only return values in the range (-π/2, π/2) and cannot distinguish angles in the second and third quadrants. To address this, most programming languages provide the atan2(y, x) function, which returns an angle in the range (-π, π] and correctly handles all four quadrants.

11. Tips for Learning Inverse Trigonometric Functions

1. Understand the principal value ranges: The range restriction of each inverse trig function is the key to understanding its behavior — memorize them thoroughly.
2. Master the complementary identities: arcsin x + arccos x = π/2 is one of the most frequently used identities in simplification and calculation.
3. Focus on graphs: Drawing the graphs of inverse trigonometric functions (by reflecting the corresponding trig function’s graph about y = x) deepens understanding of domain, range, and monotonicity.
4. Watch for composition conditions: arcsin(sin x) = x only when x ∈ [-π/2, π/2]. Outside this range, use the periodicity and symmetry of trigonometric functions to simplify.
5. Use tools: Leverage our Inverse Trigonometric Calculator to verify your calculations and improve efficiency.

Conclusion

Inverse trigonometric functions are an indispensable part of the trigonometric function family. They transform the problem of “given a ratio, find the angle” into standardized mathematical operations, with extensive and profound applications in calculus, physics, engineering, and computer science. From the refraction of light to robot motion control, from circuit analysis to computer graphics, inverse trigonometric functions are everywhere.

We hope this article has helped you build a systematic understanding of inverse trigonometric functions. Whenever you need to perform inverse trigonometric calculations for study or work, feel free to use our Inverse Trigonometric Calculator for quick and accurate results.