From triangles to waves
Trigonometry starts with right triangles and ends up describing every wave you'll ever meet — sound, light, AC current, ocean swells. Three ratios, sin, cos and tan, do all the heavy lifting.
SOHCAHTOA
For a right triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. The mnemonic SOH-CAH-TOA spells out the three ratios.
The unit circle
A circle of radius 1 centered at the origin. Any angle θ from the positive x-axis lands on the point (cos θ, sin θ). The unit circle extends sin and cos to all angles, not just acute ones.
Radians
An angle whose arc length equals its radius is 1 radian. 2π rad = 360°; π rad = 180°; π/2 rad = 90°.
Identities
The big one: sin²θ + cos²θ = 1, derived directly from Pythagoras on the unit circle.
Special angles
Hands-on tools
Drag the unit circle, transform a wave, or solve a right triangle.
Drag the unit circle
Drag the dot — angle, sin, cos, tan all update live.
All values come straight from the (x, y) coordinate of the dot.
Wave lab — y = A·sin(B(x − C)) + D
Slide A (amplitude), B (frequency), C (phase shift), D (vertical shift).
SOHCAHTOA solver
Provide any 2 known sides or 1 side + 1 acute angle.
Angle of elevation
An observer stands a distance from a tower. How tall does the angle make it?
Quiz
Flashcards
Tap to flip. ← / → keys to navigate.
Daily challenge
A new problem every day. Same problem for everyone, worldwide. Build a streak — one shot per day.
For teachers
Print-ready worksheet, answer key, teaching tips and standards alignment.
Teaching tips
Standards alignment
Reference
Formula sheet
Photo gallery
Images sourced from Wikipedia.