Sine Calculator
Calculate sine function sin(θ) and inverse sine arcsin(x) step-by-step for angles in degrees or radians.
Sine & Inverse Sine Evaluation Engine
Select calculation mode and enter input parameters.
Sine Evaluation Output
Quick Summary
Our sine calculator sin angle ratio degrees radians steps tool computes the sine ratio $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ and inverse sine $\arcsin(x)$ for any angle in degrees or radians. It displays exact radical forms, cosecant reciprocals, and unit circle Y-coordinates.
How It Works: Right-Triangle Ratios & Unit Circle
The sine function measures the vertical Y-displacement ratio of an angle:
1. **Right-Triangle Ratio:** $\sin\theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$.
2. **Unit Circle Definition:** $\sin\theta$ equals the vertical Y-coordinate of a point on a unit circle ($r=1$).
3. **Inverse Sine ($\arcsin(x)$):** Given ratio $x \in [-1, 1]$, computes principal angle $\theta$ in range $[-90^\circ, 90^\circ]$.
4. **Law of Sines:** $\a / \sin A = \b / \sin B = \c / \sin C = 2R$ (where $R$ is circumradius).
5. **Taylor Maclaurin Power Series:** $\sin x = x - \x^3 / 3! + \x^5 / 5! - \x^7 / 7! + \dots$.
Formula Explanation
Your sine calculations follow fundamental geometric and power series formulas:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for evaluating $\sin(30^\circ)$:
- Step 1 (Convert Angle to Radians): $\theta = 30^\circ \times \\pi / 180^\circ = \\pi / 6 \approx \mathbf{0.5236 \text{ rad}}$.
- Step 2 (Apply Opposite/Hypotenuse Ratio): For a 30-60-90 right triangle with opposite side 1 and hypotenuse 2: $\sin(30^\circ) = \1 / 2 = \mathbf{0.5000}$.
- Step 3 (Calculate Cosecant Reciprocal): $\csc(30^\circ) = \1 / \sin(30^\circ) = \1 / 0.5 = \mathbf{2.0000}$.
- Step 4 (Locate Unit Circle Coordinates): $(x, y) = (\cos(30^\circ), \sin(30^\circ)) = (\frac{\sqrt{3}}{2}, \1 / 2) \approx \mathbf{(0.8660, 0.5000)}$.
- Step 5 (Compute Law of Sines Multiplier): $\frac{\text{Side}}{\sin(30^\circ)} = \frac{\text{Side}}{0.5} = \mathbf{2 \times \text{Side}}$, delivering **sin(30°) = 0.5, csc(30°) = 2**!
Calculation Examples: Real-World Scenario Comparison
Compare sine angles, inverse arcsin inputs, exact radical forms, and cosecant values:
| Calculation Input | Sine Value sin(θ) / Angle θ | Exact Radical Form | Cosecant csc(θ) | Unit Circle Point (x, y) |
|---|---|---|---|---|
| 30° (π/6) | 0.5000 | 1/2 | 2.0000 | (0.8660, 0.5000) |
| 45° (π/4) | 0.7071 | √2 / 2 | 1.4142 | (0.7071, 0.7071) |
| arcsin(0.8660) | 60.00° (1.0472 rad) | π/3 rad | Ratio x = 0.8660 | (0.5000, 0.8660) |
| 240° (4π/3) | -0.8660 | -√3 / 2 | -1.1547 | (-0.5000, -0.8660) |
Benefits of Using the Sine Calculator
Utilizing this calculator provides essential physics projectile motion, AC electrical wave engineering, and acoustic sound synthesis advantages:
- Physics Projectile Motion Height & Range: Calculates vertical velocity components ($v_y = v_0 \sin\theta$) and max altitude ($h = \v_0^2 \sin^2\theta / 2g$).
- AC Electrical Engineering Alternating Current: Computes instantaneous voltage waveforms $v(t) = V_{peak} \sin(2\pi f t)$.
- Acoustic Sound Wave Frequency Synthesis: Generates pure audio tone frequencies using sinusoidal wave functions.
- Law of Sines Triangle Solver: Solves oblique triangle side lengths and unknown angles.
Frequently Asked Questions (FAQ)
What is sine (sin) in trigonometry?
Sine is the trigonometric function of an angle defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
What is the range of the sine function?
The range of sin(θ) for real angles θ is strictly between -1 and 1 ([-1, 1]).
What is inverse sine (arcsin / sin⁻¹)?
Inverse sine (arcsin) is the inverse function of sine that takes a ratio x between -1 and 1 and returns the principal angle θ in range [-90°, 90°].
Why does arcsin(2) give a domain error?
Because the maximum possible ratio for opposite over hypotenuse is 1 (or -1 for negative coordinates); ratios greater than 1 do not exist for real angles.
What is sin(0°), sin(30°), sin(45°), sin(60°), sin(90°)?
sin(0°) = 0, sin(30°) = 1/2 (0.5), sin(45°) = √2/2 (≈ 0.7071), sin(60°) = √3/2 (≈ 0.8660), sin(90°) = 1.
Is sine an even or odd function?
Sine is an ODD function: sin(-θ) = -sin(θ).
What is the period of the sine function?
The period of sin(θ) is 360° or 2π radians (sin(θ + 2π) = sin(θ)).
What is the reciprocal of sine?
The reciprocal of sine is cosecant: csc(θ) = 1 / sin(θ).
How does Law of Sines work?
Law of Sines states that for any triangle: a / sin(A) = b / sin(B) = c / sin(C).
How does a computer calculate sine without a protractor?
Computers use CORDIC algorithms or Taylor polynomial power series approximations (sin x = x - x^3/6 + x^5/120 - ...).