The Unit Circle

SOH-CAH-TOA only works inside a right triangle. An angle of 120° has no right triangle to sit in.

Put the triangle in a circle

On a circle of radius 1, the hypotenuse is 1, so cosine is just the x-coordinate and sine the y.

Now the angle can be anything

A terminal point exists for every angle, so sine and cosine are defined for all of them.

There is a second way to measure angles.

Radian is the angle whose arc is as long as the radius. A full turn is 2π radians.

The ratios freed from the triangle

On a circle of radius 1, a point at angle θ has coordinates (cos θ, sin θ). This defines the functions for every angle, including obtuse, reflex and negative ones, which the right-triangle definition cannot handle.

Signs by quadrant

Both positive in the first quadrant; only sine in the second; only tangent in the third; only cosine in the fourth. The signs come directly from the signs of the x and y coordinates, so they need no separate memorising.

Going round again repeats

Adding 360° or 2π returns to the same point, so sine and cosine are periodic. That periodicity is why they model anything that repeats — tides, sound, alternating current, seasons.

Tangent as a slope

tan θ = sin θ / cos θ is the slope of the line from the origin at angle θ. It is undefined where cos θ = 0, which is exactly where that line is vertical and has no slope.