EASA / CAR Part-66 · Module 1 · 1.3 Geometry · Cat B1/B2

The Unit Circle & Angles Beyond 90°

Objective: a right triangle only reaches 90°, but a rotating radius keeps going. Sweep the angle right round and watch the sine and cosine change sign quadrant by quadrant, tracing out the waves as they go.

Rotating radius on the unit circle

The wave it traces

Angle θ 45 °

sin θ

cos θ

Quadrant

In radians

on a unit circle: cos θ is the x coordinate, sin θ is the y coordinate · sin²θ + cos²θ = 1

Sine and cosine are coordinates, not just ratios. Put the radius of a circle of radius 1 at angle θ and its tip sits at (cos θ, sin θ). Everything about the signs follows from which way the tip has moved, left or right of centre, above or below it.

The signs follow the quadrant. In the first quadrant both are positive; in the second the tip is left of centre so cosine goes negative; in the third both are negative; in the fourth it is back on the right but below the middle, so sine is negative. That is the CAST rule, and it is simply a description of the picture.

Because the radius is 1, Pythagoras gives sin² + cos² = 1 at every angle without exception. The readout shows it holding as you sweep, which is a useful check that you have the two values the right way round.

Try this: Sweep past 90°. Which one changes sign first, and where is the tip of the radius?
Try this: Find the two angles where sine is zero, and the two where cosine is zero. What is the pattern?
Try this: Stop at 180°, 270° and 360° and read the signs. Do they match CAST?