A sine is a trigonometric function that takes an angle as input and returns a number between −1 and 1 describing how “vertical” that angle is on a circle or triangle.

Basic idea

  • In a right triangle, the sine of an angle is the ratio
    sin⁡(θ)=opposite sidehypotenuse\sin(\theta)=\dfrac{\text{opposite side}}{\text{hypotenuse}}sin(θ)=hypotenuseopposite side​.
  • On the unit circle (a circle of radius 1), sin⁡(θ)\sin(\theta)sin(θ) is the y‑coordinate of the point you reach when you rotate by angle θ\theta θ.

Key properties

  • Values of sine are always between −1 and 1, because the opposite side can never be longer than the hypotenuse in a right triangle.
  • The sine function is periodic with period 2π2\pi 2π, meaning its wave‑like graph repeats every 2π2\pi 2π radians.

Why sine matters

  • Sine is one of the main tools in trigonometry for finding unknown sides and angles in right triangles.
  • Its wave shape models many real‑world oscillations, like sound waves, light waves, and simple vibrations in physics and engineering.

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