Sine Of Angle⚠ unverified
Trigonometry / Unit Circle · Sine of an angle
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| theta | θ | degree | 30 | Angle |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| s | sin θ | — | Sine |
The science & history
Understanding the Parameters
- θ (degree) — Angle.
- Output sin θ — Sine.
How to Calculate
- Enter Angle as
theta(default 30 degree). Use the unit menu when you need a different unit. - Click Calculate. The card evaluates $\sin\theta$ and shows the result in the declared output unit; Result in (when present) converts that number.
The Science: A Rigorous Derivation (Approaching a Proof)
On the unit circle the $y$-coordinate of the ray at angle $\theta$. For acute $\theta$ this is opp/hyp.
Place an angle $\theta$ at the origin of the coordinate plane; the ray meets the unit circle at $(\cos\theta,\sin\theta)$. This definition extends the right-triangle ratios to all real $\theta$ (and, via $e^{i\theta}$, to the complex numbers): sine and cosine become the odd/even $2\pi$-periodic solutions of $y''+y=0$ with $y(0)=0$ or $1$. Signs follow the quadrant; $\tan\theta = \sin\theta/\cos\theta$ has period $\pi$.
Dimensional check. The declared output unit is dimensionless; ToolBox evaluates the
formula in SI (radians internally for every trigonometric call) and python-calc converts to
the unit on the card.
History and Development
Euler's Introductio in analysin infinitorum (1748) puts $\sin$ and $\cos$ on the unit circle and writes $e^{i\theta} = \cos\theta + i\sin\theta$. The analytic definition supersedes the triangle, which remains the computational picture for acute angles.
Related Concepts: Cosine Of Angle, Tangent Of Angle, Cosecant Of Angle, Secant Of Angle, Cotangent Of Angle