SOH CAH TOA to Unit Circle Explorer

Scrub, drag, and scale math primitives to see how they wrap into the full unit circle coordinate definitions.

Syllabus 01

SOH CAH TOA & The Right Triangle

Trigonometry begins with a simple right-angled triangle. Before circles, trig is defined purely by ratios of side lengths relative to an angle θ. Scrub the sliders or drag the top vertex directly to observe how the ratios remain consistent as sides scale.

Angle θ 35.0°
Hypotenuse (Scale) 180 px

SOH Sine Definition

sin(θ) =
Opposite Hypotenuse
=
103.2 180.0
= 0.5736

CAH Cosine Definition

cos(θ) =
Adjacent Hypotenuse
=
147.4 180.0
= 0.8192

TOA Tangent Definition

tan(θ) =
Opposite Adjacent
=
103.2 147.4
= 0.7002
Interactive: You can click and drag the top corner node of the triangle in the canvas directly!
Syllabus 02

The Bridge: Scaling Hypotenuse to 1

Why does SOH CAH TOA turn into coordinates on a circle? Because we scale down the triangle until its hypotenuse is exactly 1. When Hypotenuse = 1, the formulas simplify: Opposite = sin(θ) and Adjacent = cos(θ). Scrub the morphing slider below to watch the hypotenuse shrink to radius 1.

Scale Morpher (Shrink Hypotenuse to 1) 0%
Angle θ 40.0°

Original Triangle (H > 1)

Side lengths are scaled by a general hypotenuse H:

Opposite = H × sin(θ) = 128.6
Adjacent = H × cos(θ) = 153.2
Hypotenuse = 200.0

Scaled Unit Triangle (H = 1)

Divide all sides by the Hypotenuse H to normalize:

Opposite =
O H
= sin(θ) = 0.6428
Adjacent =
A H
= cos(θ) = 0.7660
Hypotenuse =
H H
= 1.0
Syllabus 03

The Unit Circle: 4 Quadrants & Signs

Now place the scaled triangle inside a circle of radius R = 1. As the angle θ rotates beyond 90°, the hypotenuse remains length 1, but coordinates go negative depending on the quadrant. Drag the radial handle or use the slider to scrub through all four quadrants.

Angle θ (Continuous) 120.0°

Circle Point Coordinates

P(x, y) = (cosθ, sinθ)
P( -0.5000 , 0.8660 )

Active Quadrant Status (ASTC)

Quadrant I (A)
sin(+) cos(+) tan(+)
Quadrant II (S)
sin(+) cos(-) tan(-)
Quadrant III (T)
sin(-) cos(-) tan(+)
Quadrant IV (C)
sin(-) cos(+) tan(-)

Remember: All, Sine, Tangent, Cosine are positive in Quadrants I, II, III, and IV respectively.

Syllabus 04

The Live Wave Generator

If we plot the y-coordinate (Sine) and x-coordinate (Cosine) of the unit circle as a function of the angle over time / space, we generate periodic waves. Drag the slider to sweep from 0° to 720° (two full periods). Watch the horizontal and vertical glowing tracer lines map coordinate positions directly onto the sine and cosine wave graphs.

Continuous Angle θ 215.0° / 3.75 rad
y = sin(θ) = -0.5736
x = cos(θ) = -0.8192
Syllabus 05

The Tangent Explorer (Slope & Limits)

Why is the tangent function called "tangent"? Geometrically, it is the length of the line segment that is tangent to the circle at point (1, 0), extended until it intersects the secant line (radial ray). Scrub the slider towards 90° and see why tangent shoots to infinity.

Tangent Angle θ 45.0°

Geometric Tangent Line

The segment of the vertical line tangent to the circle at $(1, 0)$ is:

tan(θ) = 1.0000

The length of the extended radial ray (Secant) is:

sec(θ) = 1.4142

Understanding the Limit at 90°

As θ → 90°, the radial ray pointing to the point on the unit circle becomes completely vertical. Since it is vertical, it runs parallel to our tangent line at x = 1. Parallel lines never intersect! Therefore, the segment length becomes infinite:

tan(89.5°) ≈ 114.6
tan(89.9°) ≈ 573.0
tan(90°) = ∞ (Undefined)