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)