Terminal Side of An Angle in Standard Position Calculator
'/%3E%3Ccircle cx='48' cy='39' r='19' fill='%23f2c9a5'/%3E%3Cpath d='M27 35c3-16 39-17 42 2-8-8-27-9-42-2z' fill='%23374151'/%3E%3Cpath d='M21 86c5-24 49-28 56 0z' fill='%232563eb'/%3E%3Ccircle cx='41' cy='41' r='2' fill='%23111827'/%3E%3Ccircle cx='55' cy='41' r='2' fill='%23111827'/%3E%3Cpath d='M42 52c4 3 9 3 13 0' stroke='%2392400e' stroke-width='3' fill='none' stroke-linecap='round'/%3E%3C/svg%3E)
Reviewed by Calculator Editorial Team
An angle in standard position is positioned with its vertex at the origin of a coordinate plane and its initial side along the positive x-axis. The terminal side is the ray that rotates from the initial side to form the angle. This calculator helps you determine the coordinates of the terminal side given an angle measurement.
What is the terminal side of an angle?
In trigonometry, an angle in standard position has its vertex at the origin (0,0) of a coordinate plane and its initial side along the positive x-axis. The terminal side is the ray that extends from the origin and forms the angle with the initial side.
The terminal side's position can be described using coordinates (x, y) on the unit circle, where the radius (r) is 1. For angles measured in standard position, the terminal side's coordinates can be calculated using trigonometric functions:
- x = r × cos(θ)
- y = r × sin(θ)
Where θ is the angle measurement and r is the radius (distance from the origin to the point on the terminal side).
How to calculate the terminal side
To find the terminal side of an angle in standard position:
- Identify the angle measurement (θ) in degrees or radians.
- Determine the radius (r) if it's not a unit circle (r = 1).
- Calculate the x-coordinate using x = r × cos(θ).
- Calculate the y-coordinate using y = r × sin(θ).
- Plot the point (x, y) on the coordinate plane.
Key Formulas
For an angle θ in standard position with radius r:
- x-coordinate: x = r × cos(θ)
- y-coordinate: y = r × sin(θ)
Using the calculator
Our interactive calculator makes it easy to find the terminal side coordinates:
- Enter the angle measurement in degrees or radians.
- Specify the radius (default is 1 for unit circle).
- Click "Calculate" to see the coordinates.
- View the result and chart visualization.
Note
The calculator uses JavaScript to perform real-time calculations. For best results, use a modern browser.
Example calculation
Let's find the terminal side coordinates for an angle of 45° with radius 2:
- x = 2 × cos(45°) ≈ 2 × 0.7071 ≈ 1.4142
- y = 2 × sin(45°) ≈ 2 × 0.7071 ≈ 1.4142
The terminal side coordinates are approximately (1.4142, 1.4142).
| Angle | Radius | x-coordinate | y-coordinate |
|---|---|---|---|
| 45° | 2 | 1.4142 | 1.4142 |
| 90° | 1 | 0 | 1 |
| 180° | 3 | -3 | 0 |
FAQ
What is the difference between standard position and other angle positions?
An angle in standard position has its vertex at the origin and initial side along the positive x-axis. Other positions may have different vertex locations or initial sides.
Can I use radians instead of degrees?
Yes, the calculator accepts both degrees and radians. Select the appropriate unit from the dropdown menu.
What if the radius is not 1?
The calculator allows you to specify any radius value. The coordinates will be scaled accordingly.
How accurate are the calculations?
The calculator uses JavaScript's Math functions, which provide accurate results up to the precision limits of floating-point arithmetic.