Sine Calculator
Calculate the sine of an angle given in degrees or radians. Our Sine Calculator is easy to use and provides instant results along with a sine wave chart.
Calculate Sine
Sine Wave Chart
Common Sine Values
| Angle (Degrees) | Angle (Radians) | Sine Value (sin θ) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 ≈ 0.5236 | 0.5 |
| 45° | π/4 ≈ 0.7854 | √2/2 ≈ 0.7071 |
| 60° | π/3 ≈ 1.0472 | √3/2 ≈ 0.8660 |
| 90° | π/2 ≈ 1.5708 | 1 |
| 180° | π ≈ 3.1416 | 0 |
| 270° | 3π/2 ≈ 4.7124 | -1 |
| 360° | 2π ≈ 6.2832 | 0 |
What is a Sine Calculator?
A Sine Calculator is a tool used to determine the sine of a given angle. The sine is a fundamental trigonometric function, one of the three primary functions along with cosine and tangent. It is defined in the context of a right-angled triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse. However, its definition extends beyond right-angled triangles through the unit circle, making it applicable to any angle, positive or negative, and even angles greater than 360 degrees or 2π radians. Our online Sine Calculator allows you to input an angle in either degrees or radians and instantly get the sine value.
Anyone working with angles, waves, oscillations, or circular motion can benefit from using a Sine Calculator. This includes students of mathematics, physics, and engineering, as well as professionals like engineers, architects, animators, and scientists.
A common misconception is that sine only applies to angles within a triangle (0 to 180 degrees). However, the sine function is periodic and defined for all real numbers, representing wave-like patterns found throughout nature and technology.
Sine Calculator Formula and Mathematical Explanation
For an acute angle θ in a right-angled triangle:
sin(θ) = Opposite Side / Hypotenuse
More generally, using the unit circle (a circle with radius 1 centered at the origin), if we draw a radius from the origin to a point (x, y) on the circle such that the radius makes an angle θ with the positive x-axis, then:
sin(θ) = y
This unit circle definition allows us to find the sine of any angle. The sine function is periodic with a period of 360° or 2π radians, meaning sin(θ) = sin(θ + 360°) = sin(θ + 2π).
If the angle is given in degrees, it's often converted to radians for calculations using the formula: Radians = Degrees × (π / 180). Our Sine Calculator handles this conversion automatically if you input the angle in degrees.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| θ (Angle) | The input angle whose sine is to be calculated | Degrees or Radians | Any real number |
| sin(θ) | The sine of the angle θ | Dimensionless ratio | -1 to 1 |
| π (Pi) | Mathematical constant, approx. 3.14159 | N/A | ~3.14159 |
Practical Examples (Real-World Use Cases)
The Sine Calculator is useful in various fields:
Example 1: Physics – Simple Harmonic Motion
An object oscillating on a spring can be described by x(t) = A * sin(ωt + φ). If the amplitude A is 10 cm, angular frequency ω is 2 rad/s, and phase φ is 0, to find the position at t = 0.5s, we calculate sin(2 * 0.5) = sin(1 radian). Using the Sine Calculator with 1 radian (approx 57.3 degrees), sin(1) ≈ 0.841. So, x(0.5) ≈ 10 * 0.841 = 8.41 cm.
Example 2: Engineering – AC Circuits
The voltage in an AC circuit is given by V(t) = Vmax * sin(ωt). If Vmax = 170V and ω = 120π rad/s (for 60Hz frequency), at t = 1/240 s, the angle is 120π * (1/240) = π/2 radians (90 degrees). sin(π/2) = 1. So, V(1/240) = 170 * 1 = 170V (peak voltage).
How to Use This Sine Calculator
Using our Sine Calculator is straightforward:
- Enter the Angle Value: Type the numerical value of the angle into the "Angle Value" input field.
- Select the Unit: Choose whether the angle you entered is in "Degrees" or "Radians" by selecting the corresponding radio button.
- View the Results: The calculator automatically updates and displays the sine of the angle, as well as the angle converted to both degrees and radians. The primary result is the sine value.
- Interpret the Chart: The sine wave chart visually represents the sine function, and a red dot marks the point corresponding to your input angle and its sine value.
- Reset (Optional): Click "Reset" to return the input angle to 30 degrees.
- Copy Results (Optional): Click "Copy Results" to copy the angle in degrees, radians, and the calculated sine value to your clipboard.
The results from the Sine Calculator give you the sine value, which can be used in further calculations or analysis depending on your needs.
Key Factors That Affect Sine Results
The main factors affecting the output of the Sine Calculator are:
- Angle Value: The numerical value of the angle is the primary input. The sine function varies between -1 and 1 as the angle changes.
- Angle Unit: Whether the angle is input in degrees or radians is crucial. sin(30 degrees) is 0.5, while sin(30 radians) is approximately -0.988. Our Sine Calculator requires you to specify the unit.
- Calculator Precision: The underlying `Math.sin()` function in JavaScript uses floating-point arithmetic, which has very high precision, but it's good to be aware that it's an approximation for most values.
- Periodicity: The sine function is periodic with a period of 360° or 2π radians. Adding or subtracting multiples of 360° (or 2π radians) to an angle will result in the same sine value.
- Quadrant of the Angle: The sign (positive or negative) of the sine value depends on the quadrant in which the angle terminates (0-90°: +, 90-180°: +, 180-270°: -, 270-360°: -).
- Input Validity: The Sine Calculator expects a numerical value for the angle. Non-numeric input will lead to an error or NaN (Not a Number) result.
Frequently Asked Questions (FAQ)
What is sine?
Sine is a trigonometric function that relates an angle of a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. More broadly, it's a periodic function describing smooth oscillations, defined using the y-coordinate on a unit circle.
What is the range of the sine function?
The sine of any real angle will always be between -1 and 1, inclusive [-1, 1].
How do I find the sine of a negative angle?
The sine function is an odd function, meaning sin(-θ) = -sin(θ). So, the sine of a negative angle is the negative of the sine of the positive angle. Our Sine Calculator handles negative angles correctly.
What is the sine of 90 degrees?
sin(90°) = 1.
What is the sine of 0 degrees?
sin(0°) = 0.
What is arcsin or sin-1?
Arcsin, or sin-1, is the inverse sine function. If sin(θ) = x, then arcsin(x) = θ. It gives you the angle whose sine is x. Note that there are infinitely many such angles, but the principal value is usually returned (between -90° and 90° or -π/2 and π/2).
Can I use this Sine Calculator for angles larger than 360 degrees?
Yes, the Sine Calculator can handle angles larger than 360 degrees (or 2π radians) because of the periodic nature of the sine function.
Does this calculator use degrees or radians?
Our Sine Calculator allows you to input the angle in either degrees or radians. You select the unit using the radio buttons.