Arccos Calculator
An arccos calculator is a tool that computes the inverse cosine of a given value. The inverse cosine function, also known as arccosine or \( \cos^{-1} \), determines the angle whose cosine equals the provided number. This function is widely used in trigonometry, geometry, and engineering calculations.
What Is Arccos?
The arccosine function is the reverse operation of the cosine function. In mathematical terms:
\[
\cos(\theta) = x \implies \arccos(x) = \theta
\]
- Input Range: \( -1 \leq x \leq 1 \)
- Output Range: \( 0 \leq \theta \leq \pi \) (in radians) or \( 0^\circ \leq \theta \leq 180^\circ \) (in degrees)
Why Use an Arccos Calculator?
The calculator simplifies the process of finding angles, especially when precision matters. Instead of manually solving equations or referring to lookup tables, users can input a value and get an immediate result.
Steps to Use the Arccos Calculator
- Input the Value: Enter a number between \( -1 \) and \( 1 \).
- Select Angle Units: Choose between radians or degrees.
- View the Result: The calculator displays the corresponding angle.
Example Calculation
Suppose you want to find the angle where \( \cos(\theta) = 0.5 \):
- Input \( x = 0.5 \).
- Choose “degrees” as the unit.
- The calculator gives \( \arccos(0.5) = 60^\circ \).
Arccos Table
Below is a table showing cosine values and their corresponding angles in radians and degrees for common reference:
| Cos(x) | Arccos(x) (Radians) | Arccos(x) (Degrees) |
|---|---|---|
| 1.0 | 0 | 0° |
| 0.866 | \( \frac{\pi}{6} \) (0.5236) | 30° |
| 0.707 | \( \frac{\pi}{4} \) (0.7854) | 45° |
| 0.5 | \( \frac{\pi}{3} \) (1.0472) | 60° |
| 0 | \( \frac{\pi}{2} \) (1.5708) | 90° |
| -0.5 | \( \frac{2\pi}{3} \) (2.0944) | 120° |
| -0.707 | \( \frac{3\pi}{4} \) (2.3562) | 135° |
| -0.866 | \( \frac{5\pi}{6} \) (2.6179) | 150° |
| -1.0 | \( \pi \) (3.1416) | 180° |
How to Use the Table
- Find the cosine value in the first column.
- Look at the corresponding angle in radians or degrees.
This table covers common angles and provides a quick reference for solving inverse cosine problems, making it an invaluable tool for trigonometric calculations.