Inverse trigonometric functions often feel confusing at first because they seem to reverse everything students learn about basic trig functions like sine, cosine, and tangent. Instead of starting with an angle and finding a ratio, an inverse trigonometric function works backward as it uses a ratio to find the angle. This idea of an inverse is central in mathematics and appears frequently in calculus, physics, geometry, and engineering problems. Since inverse trig functions involve new notation, restricted domains, and calculator settings, it’s easy to mix them up with regular trig functions.
In this article, you’ll learn how inverse trig functions work, why they are considered inverse functions, and how to use them correctly. We’ll walk through clear definitions, simple examples, common formulas, and mistakes to avoid so you can confidently solve problems involving inverse trigonometric functions.
What Are Inverse Trigonometric Functions?
An inverse trigonometric function is a type of inverse function that reverses the action of the basic trig functions sine, cosine, and tangent. While regular trig functions take an angle θ and return a ratio, inverse trig functions do the opposite: they take a ratio and find the angle that produces it. For example, instead of computing sinθ, the inverse sine function finds the angle whose sine equals a given value.
The most common inverse trig functions are written using special notation:
A key point is that −1(x) does not mean 1/sin(x). It means the inverse of the sine function. Because trig functions are not one-to-one over all real numbers, their domains are restricted so the inverse exists and returns a single, principal value. These inverse trig functions are widely used in calculus, especially when working with derivatives, graphs, and solving equations involving angles.
The Three Main Inverse Trigonometric Functions
The most commonly used inverse trigonometric functions are the inverse sine, inverse cosine, and inverse tangent. Each one is an inverse function of a basic trig function and is designed to find an angle θ\thetaθ from a known trigonometric ratio. Because trig functions are not one-to-one over all real numbers, their domains are restricted so the inverse returns a single, meaningful value—this is especially important in calculus, graphing, and when working with derivatives.
Inverse Sine (arcsin)
The inverse sine function, written as arcsin(x) or sin−1(x), finds the angle whose sine equals a given value. Since the sine function only produces outputs between –1 and 1, the input range of arcsin is limited to that interval. Its output range (principal values) is restricted to −𝜋/2≤𝑦≤𝜋/2, ensuring the inverse sine function is one-to-one.
Inverse Cosine (arccos)
The inverse cosine function, written as arccos(𝑥) or cos−1(x), returns the angle whose cosine equals a given number. Like sine, cosine functions also output values between –1 and 1, so the input range is the same. However, the output range is 0≤y≤π, which aligns with standard unit circle conventions.
Inverse Tangent (arctan)
The inverse tangent function, written as arctan(𝑥) or
tan−1(x), finds the angle whose tangent equals a given value. Unlike sine and cosine, the tangent function accepts all real numbers as inputs. Its output range is restricted to
−𝜋/2<𝑦<𝜋/2, excluding values where tangent is undefined.
Domain and Range Summary
| Inverse Trig Function | Input (Domain) | Output (Range) |
|---|---|---|
| arcsin(x) | −1 ≤ x ≤ 1 | −π/2 ≤ y ≤ π/2 |
| arccos(x) | −1 ≤ x ≤ 1 | 0 ≤ y ≤ π |
| arctan(x) | All real numbers | −π/2 < y < π/2 |
How to Do Inverse Trigonometric Functions Step by Step
Solving problems with an inverse trigonometric function becomes much easier when you follow a clear, consistent process. Since inverse trig functions work as an inverse function of the original trig functions, the goal is always to find the angle 𝜃 that produces a given trigonometric value. This step-by-step method works well for algebra, calculus, and graph-based problems
Step 1: Identify the Inverse Function
First, determine which inverse trig function you are working with:
- arcsin(sin−1) → inverse sine
- arccos(cos−1) → inverse cosine
- arctan(tan−1) → inverse tangent
Each inverse corresponds to a specific trig function—sine, cosine, or the tangent function—so identifying it correctly is essential before applying any formula or notation.
Step 2: Understand What Value Is Given
Next, look at the value inside the inverse function. It may be written as a ratio (such as
1/2) or as a decimal. If the value matches a common trigonometric ratio from the unit circle, you may not need a calculator. Otherwise, using a calculator is appropriate, especially in calculus problems involving derivatives.
Step 3: Find the Angle
Now determine the angle
𝜃 whose sine, cosine, or tangent equals the given value.
- Using known values: For example, since sin 30∘ = 1/2, then arcsin (1/2) = 𝜋/6.
- Using a calculator: Make sure your calculator is set to degree or radian mode as required. Most calculus problems use radians, where answers are expressed in terms of 𝜋.
Step 4: Check the Domain and Range
Finally, verify that your answer lies within the principal range of the inverse function. Because inverse trig functions require restricted domains to remain one-to-one, the final angle must satisfy the correct range (such as −𝜋/2≤𝑦≤𝜋/2 for inverse sine). This step ensures your solution is valid and consistent with the definition of the inverse.
Worked Examples
The best way to understand inverse trigonometric functions is to see them in action. In each example below, the goal is to use an inverse trig function to find the angle 𝜃 that produces a given trigonometric value. All answers will be expressed in radians, which is the standard unit used in calculus unless stated otherwise.
These examples show how inverse trig functions reverse the original trig functions by converting known ratios into angles.
When Are Inverse Trigonometric Functions Used?
An inverse trigonometric function is used whenever you need to work backward from a trigonometric ratio to find an angle. In geometry, inverse trig functions help determine unknown angles in triangles when side lengths or ratios are known. In physics, they are commonly applied to problems involving motion, forces, and vectors, where angles must be calculated from sine, cosine, or tangent values.
In engineering and navigation, inverse trig functions support direction finding, slope calculations, and trajectory analysis. They also play an important role in calculus, especially when evaluating integrals and working with derivatives of inverse functions. Anytime a problem requires reversing standard trig functions using a precise formula, inverse trig functions provide the solution.
Conclusion
Inverse trigonometric functions are powerful tools that allow you to work backward from trig functions to find angles. Instead of starting with an angle and computing sine, cosine, or tangent, an inverse trigonometric function uses a known ratio to determine the correct value of 𝜃. While inverse trig functions may feel confusing at first especially with new notation, restricted domains, and radians like 𝜋 this is completely normal.
With regular practice, the patterns become much clearer. To build confidence, try solving more examples, sketching the graph of the inverse, and exploring how inverse functions appear in calculus, particularly in derivatives and integrals. Mastery comes with repetition and application.
