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Calculate the sine of any angle in degrees or radians. Sine represents the y-coordinate on the unit circle and is fundamental to trigonometry.
| Degrees | Radians | Exact Value | Decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0.0000 |
| 30° | π/6 | 1/2 | 0.5000 |
| 45° | π/4 | √2/2 | 0.7071 |
| 60° | π/3 | √3/2 | 0.8660 |
| 90° | π/2 | 1 | 1.0000 |
| 120° | 2π/3 | √3/2 | 0.8660 |
| 135° | 3π/4 | √2/2 | 0.7071 |
| 150° | 5π/6 | 1/2 | 0.5000 |
| 180° | π | 0 | 0.0000 |
| 270° | 3π/2 | -1 | -1.0000 |
| 360° | 2π | 0 | -0.0000 |
Domain: All real numbers (-∞, ∞)
Range: [-1, 1]
Period: 2π radians (360°)
sin(θ + 2π) = sin(θ)
Odd function: sin(-θ) = -sin(θ)
Origin symmetry
Zeros: 0°, 180°, 360°, ...
Max: 90°, Min: 270°
The sine function is one of the six fundamental trigonometric functions. It originated from the study of right triangles and circular motion, dating back to ancient civilizations including the Babylonians, Greeks, and Indians. The modern sine function is defined using the unit circle.
In a right triangle, the sine of an angle θ equals the ratio of the length of the side opposite the angle to the length of the hypotenuse: sin(θ) = opposite / hypotenuse.
On the unit circle (a circle with radius 1 centered at the origin), the sine of an angle equals the y-coordinate of the point where the terminal side of the angle intersects the circle. This geometric interpretation extends the definition to all real numbers, not just acute angles.
The sine function can be calculated using the unit circle definition or various series expansions. For common angles, memorizing exact values is helpful.
Step 1: Convert to radians if needed: 30° = π/6 rad
Step 2: Recall that sin(30°) = 1/2 (exact value)
Result: sin(30°) = 0.5
Step 1: Convert to radians: 45° = π/4 rad
Step 2: Recall that sin(45°) = √2/2 (exact value)
Result: sin(45°) ≈ 0.7071
Step 1: 135° is in Quadrant II
Step 2: Reference angle = 180° - 135° = 45°
Step 3: Sine is positive in Quadrant II
Result: sin(135°) = sin(45°) = √2/2 ≈ 0.7071
Sine waves model sound waves, light waves, ocean waves, and alternating current electricity. Understanding sine is essential for physics and engineering.
Sine functions help calculate distances, heights, and positions in navigation, astronomy, and land surveying applications.
Sine and cosine are used to rotate objects, create smooth animations, and generate circular and curved paths in 2D and 3D graphics.
Fourier analysis decomposes complex signals into sine waves, enabling compression, filtering, and analysis of audio, images, and data.
All Students Take Calculus: All trig functions positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4.
Always check if your calculator is in degree or radian mode. sin(30) in radians is NOT the same as sin(30°).
Learn sine values for 0°, 30°, 45°, 60°, and 90°. You can derive all other values using reference angles.
For any angle, find its reference angle (acute angle to x-axis) and apply the appropriate sign based on the quadrant.
Sine is a trigonometric function that returns the y-coordinate of a point on the unit circle. In a right triangle, sin(θ) = opposite/hypotenuse.
Because sin(-θ) = -sin(θ). Negating the angle reflects the point across the x-axis, which negates the y-coordinate.
Sine is positive in Quadrants I and II (0° to 180°), and negative in Quadrants III and IV (180° to 360°).
The range of sine is [-1, 1]. The sine of any angle will always be between -1 and 1, inclusive.
The period of sine is 2π radians (360°). This means sin(θ + 2π) = sin(θ) for all values of θ.
The formula is the same, but you must convert degrees to radians first. To convert: radians = degrees × (π/180).
Sine equals zero at 0°, 180°, 360°, and all integer multiples of 180° (or 0, π, 2π, ... in radians).
On the unit circle, the sine of an angle equals the y-coordinate of the point where the terminal side of the angle intersects the circle.