How to Calculate Inverse Cosine on Calculator and with Formulas
The concept of Inverse Cosine is a crucial part of trigonometry. It helps us find the angle when a cosine value is given and appears frequently in board exams, JEE, NEET, and real-life applications like physics and engineering.
What Is Inverse Cosine?
Inverse Cosine, commonly written as arccos or cos⁻¹(x), is the function used to find the angle whose cosine value is a specific number. For example, if cos(θ) = x, then θ = cos⁻¹(x). This is not to be confused with the reciprocal of cosine (which is secant or sec). You’ll see inverse cosine used whenever you need to reverse a cosine calculation, such as in the trigonometric functions chapter, in vector direction problems in physics, and also in geometry tasks.
Key Formula for Inverse Cosine
Here’s the standard formula: \( \cos^{-1}(x) = \theta \quad \text{if and only if} \quad \cos(\theta) = x \), where \( x \) is between –1 and 1.
| Notation | Meaning |
|---|---|
| arccos(x) | Inverse cosine of x |
| cos⁻¹(x) | Angle whose cosine is x |
Domain and Range of Inverse Cosine
It’s important to remember that cos⁻¹(x) is defined only for x values between –1 and 1.
| Domain | Range (Radians) | Range (Degrees) |
|---|---|---|
| –1 ≤ x ≤ 1 | 0 ≤ θ ≤ π | 0° ≤ θ ≤ 180° |
How to Calculate Inverse Cosine (cos⁻¹x): Step-by-Step
- Check if the value of x is between –1 and 1.
Example: x = 1/2 (valid), x = 2 (not valid)
- On your calculator, press the ‘SHIFT’ or ‘2nd’ key, then ‘COS’ to access cos⁻¹.
- Enter the value of x and select the desired mode (degrees or radians).
- Press ‘=’ and read the angle. For cos⁻¹(1/2): Most calculators show 60°, which can also be written as π/3 radians.
- Negative values return angles greater than 90° and up to 180°.
Common Values for cos⁻¹(x)
| x | cos⁻¹(x) in Degrees | cos⁻¹(x) in Radians |
|---|---|---|
| 1 | 0° | 0 |
| 1/2 | 60° | π/3 |
| 0 | 90° | π/2 |
| –1/2 | 120° | 2π/3 |
| –1 | 180° | π |
Cross-Disciplinary Usage
Inverse Cosine is not only useful in Maths but also plays a crucial role in Physics (finding angles, resolving forces), Computer Science (graphics/animation), and engineering (signal resolution, navigation). Students preparing for JEE and NEET will encounter problems requiring cos⁻¹ frequently.
Step-by-Step Illustration: Solved Example
Example: Find the angle θ such that cos θ = 0.
1. We need to find θ for which cos θ = 0.2. θ = cos⁻¹(0).
3. From the value table, cos⁻¹(0) = 90° or π/2 radians.
4. Final answer: θ = 90° (or π/2 radians).
Speed Trick or Shortcut
Here’s a trick: For quick calculation of cos⁻¹(special values), remember the standard values (such as 0, ±1/2, ±1). These are often used in MCQs. For cos⁻¹(–1/2), directly recall it is 120° (2π/3 radians) for the principal value.
Tricks like this are practical for exams. Vedantu’s live classes share more such shortcut methods for trigonometric inverses.
Try These Yourself
- Find cos⁻¹(–1).
- Calculate the angle whose cosine is 1/2.
- What happens if you try cos⁻¹(2)?
- Find all angles θ in [0°, 180°] for which cos θ = 0.
Frequent Errors and Misunderstandings
- Confusing inverse cosine (cos⁻¹) with cosine reciprocal (sec). Remember, sec(x) = 1/cos(x), NOT cos⁻¹(x).
- Trying to calculate cos⁻¹(x) for x values outside –1 to 1.
- Forgetting that calculator returns only the principal value (within [0°, 180°]).
Relation to Other Concepts
Inverse Cosine is related closely to sin inverse (arcsin) and tan inverse (arctan). Understanding cos⁻¹(x) will make it easier to deal with trigonometric identities and equations. For standard values, refer to the trigonometry table and trigonometric identities page.
Classroom Tip
To quickly remember the domain and range: The input for cos⁻¹(x) must be between –1 and 1, and the output is always an angle between 0° and 180°. Vedantu’s teachers recommend memorizing the “cosine curve” on the unit circle so you always know which angles to expect.
We explored Inverse Cosine—from its definition, formula, domain/range, key values, mistakes, and close links with other trig functions. Continue learning and practicing with Vedantu for more shortcuts and expert guidance in trigonometry!
For further reading, visit: Trigonometric Functions | Trigonometric Identities | Sin Inverse (arcsin) in Maths | Trigonometry Table | Tan Inverse (arctan) in Maths
FAQs on Inverse Cosine (arccos or cos⁻¹) – Definition, Graph, Formula & Examples
1. What is the inverse cosine function, often written as arccos(x) or cos⁻¹(x)?
The inverse cosine function, arccos(x) or cos⁻¹(x), is used to find the angle whose cosine is a specific value 'x'. For example, if cos(θ) = x, then θ = cos⁻¹(x). It essentially “reverses” the cosine operation to find the original angle, which is a fundamental concept for solving trigonometric equations in the CBSE 2025-26 syllabus.
2. What are the official domain and range of the inverse cosine function?
For the function y = cos⁻¹(x), it is defined with a specific domain and range to ensure it has a unique output for each input.
- The domain (possible input values for 'x') is [-1, 1]. You cannot find the inverse cosine of a number greater than 1 or less than -1.
- The range (the output angle 'y') is [0, π] or [0°, 180°]. This is known as the principal value branch for inverse cosine.
3. What is a simple example of calculating an inverse cosine value?
Let's find the value of cos⁻¹(1/2). We are looking for the angle θ in the principal value range [0, π] such that cos(θ) = 1/2. From our knowledge of standard trigonometric values, we know that cos(π/3) or cos(60°) is 1/2. Since π/3 is within the required range, the answer is cos⁻¹(1/2) = π/3 radians or 60°.
4. How does the graph of inverse cosine, y = cos⁻¹(x), differ from the graph of y = cos(x)?
The graphs of y = cos(x) and y = cos⁻¹(x) are fundamentally different.
- y = cos(x) is a periodic wave that oscillates between -1 and 1 along the y-axis, with its domain being all real numbers.
- y = cos⁻¹(x) is not a wave. It is a function segment that exists only for x-values from -1 to 1 (its domain) and gives angle values from 0 to π (its range). Visually, the graph of y = cos⁻¹(x) is a reflection of the y = cos(x) graph (when restricted to the [0, π] interval) across the line y = x.
5. What is the most common mistake students make with the cos⁻¹(x) notation?
A very common mistake is confusing the inverse cosine function, cos⁻¹(x), with the reciprocal of cosine, (cos(x))⁻¹. It is crucial to understand the difference:
- cos⁻¹(x) or arccos(x) is the inverse function that finds an angle.
- (cos(x))⁻¹ is equal to 1/cos(x), which is the secant function, sec(x). These two are completely different concepts with different mathematical purposes.
6. Why is the range of inverse cosine restricted to [0, π]?
The range of y = cos⁻¹(x) is restricted to [0, π] to ensure that it is a true function, meaning each input 'x' has only one unique output 'y'. The original cosine function y = cos(x) is many-to-one (e.g., cos(π/3) and cos(5π/3) both equal 0.5). By restricting the output of the inverse to the [0, π] interval, we select a single, standard angle, known as the principal value, making the inverse relationship predictable and consistent for all calculations.
7. Where is the concept of inverse cosine applied in other subjects like Physics?
The inverse cosine function has important real-world applications beyond just mathematics. In Physics, it is crucial for:
- Vector Analysis: Finding the angle between two vectors using the dot product formula.
- Resolving Forces: Determining the direction of a resultant force or the angle of an object on an inclined plane.
- Optics: Calculating angles of refraction and reflection in certain scenarios involving Snell's law.
