What is a unit circle in trigonometry?

A unit circle is a circle of radius 1 centered at the origin (0,0) in the Cartesian coordinate system. It is used to define trigonometric functions for all real numbers.

Why does cosine represent the X-coordinate?

Since the hypotenuse on the unit circle is 1, the ratio cos(θ) = adjacent / hypotenuse simplifies to adjacent / 1. Hence, the horizontal adjacent length (X-coordinate) is exactly cos(θ).

Which trigonometric ratios are undefined at 90 degrees?

At 90° (π/2 radians), the X-coordinate is 0. Since tangent (y/x) and secant (1/x) require division by x, they are undefined at 90°.

Unit Circle Explorer - Trigonometric Ratios Visualizer

Trig Ratios R = 1

sin(θ) = y0.7071

cos(θ) = x0.7071

tan(θ) = y/x 1.0

sec(θ) = 1/x 1.4142

csc(θ) = 1/y 1.4142

cot(θ) = x/y 1.0

Variable Adjuster

Angle in Degrees (θ)45°

0360

Auto-Sweep Engine

Target Param:

Sweep Speed: 2x

Unit Circle Explorer

UNCIR

The unit circle is a circle of radius 1 centered at the coordinate origin. Tying trigonometry to coordinates on the unit circle lets us extend trigonometric ratios to any angle, defining sine as Y and cosine as X.

\begin{aligned}x^2 + y^2 = 1 \\ \cos^2\theta + \sin^2\theta = 1\end{aligned}

Whiteboard Solver Steps

Step 1

Angle Conversion

We begin by converting the input angle from degrees ( $45^\circ$ ) to radians, which is the standard angular unit in calculus and physics. A full circle is $360^\circ$ or $2\pi$ radians.

\theta = 45^\circ = \theta_{rad} = \theta \times \frac{\pi}{180^\circ} \approx 0.7854\text{ rad}

Step 2

Identify Unit Circle Coordinates (x, y)

On a coordinate grid, a Unit Circle has a radius of $R = 1$ centered at $(0, 0)$ . Any angle $\theta$ sweeps out a radial vector that intersects the circle boundary at coordinate $P(x, y)$ . By definition, the $x$ -coordinate is the cosine of the angle ( $x = \cos\theta \approx 0.7071$ ) and the $y$ -coordinate is the sine of the angle ( $y = \sin\theta \approx 0.7071$ ).

P(x, y) = (\cos\theta, \sin\theta) \approx (0.7071, 0.7071)

Step 3

Calculate Primary Trigonometric Ratios

The primary trigonometric ratios are computed directly from the coordinate values: Sine represents vertical displacement, Cosine represents horizontal displacement, and Tangent represents the slope of the radial line ( $y/x$ ). At $90^\circ$ and $270^\circ$ , $x = 0$ , causing Tangent to approach infinity (undefined).

\begin{aligned}\sin\theta = y \approx 0.7071 \\ \cos\theta = x \approx 0.7071 \\ \tan\theta = \frac{y}{x} = 1.0000\end{aligned}

Step 4

Calculate Reciprocal Ratios

The reciprocal ratios are defined as the inverse of the primary functions: Cosecant ( $1/\sin$ ), Secant ( $1/\cos$ ), and Cotangent ( $1/\tan$ ). These values swell to infinity when their respective denominators approach zero.

\begin{aligned}\csc\theta = \frac{1}{y} = 1.4142 \\ \sec\theta = \frac{1}{x} = 1.4142 \\ \cot\theta = \frac{x}{y} = 1.0000\end{aligned}

RV Anveshana