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Trigonometric Functions

Radian measure, the unit circle, and periodic graphs.

Interactive graph

Interactive Lab

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Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualInstead of degrees, angles are measured by the arc length they sweep on the unit circle. A full circle is 2π.

Algebra $\pi \text{ rad} = 180^\circ$ . $2\pi \text{ rad} = 360^\circ$ .

2\pi = 360^\circ

VisualA circle of radius 1. The x-coordinate of a point is $\cos(\theta)$ , the y-coordinate is $\sin(\theta)$ .

Algebra $x^2 + y^2 = 1$ . Coordinates: $(\cos\theta, \sin\theta)$ .

\cos^2\theta + \sin^2\theta = 1

VisualSine and cosine repeat their patterns endlessly. The basic cycle repeats every $2\pi$ .

AlgebraPeriod of $\sin(x)$ and $\cos(x)$ is $2\pi$ .

f(x + 2\pi) = f(x)

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

Core unit circle identity

Conversion

\pi \text{ rad} = 180^\circ

Degrees to radians

Period

T = 2\pi

For basic sin/cos

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easytrigonometry

Convert

\pi

radians to degrees.

easytrigonometry

What is the period of

y = \sin(x)

in radians?