Trigonometry A — Triangles & the Unit Circle

The unit circle

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: triangles only handle angles up to 90°. To go all the way around, trig moves to the unit circle — a circle of radius 1 where a point at angle φ has coordinates (cos φ, sin φ). Now sine and cosine are defined for every angle, and one identity holds always — it's just Pythagoras on that radius-1 triangle: sin²φ + cos²φ = 1.

\phi := 132 deg

= 132 deg any angle you like — even past 90°

✓ pass

\mathrm{abs}\left(\mathrm{sin}\left(\phi\right)^{2} + \mathrm{cos}\left(\phi\right)^{2} - 1\right) < 10^{-9}

sin² + cos² = 1, for every angle

sqrt(1 - x^2)

upper half of the unit circle x² + y² = 1

Real-world hook: the unit circle is the math behind rotation — spinning a sprite in a game, orienting a robot arm, or tracking the angle of a Ferris wheel car as it goes all the way around and past the top.

Try it yourself: what angle (in degrees), between 0° and 90°, has a sine of 1/2?

ang_{you} = 30

✏️ Your turn: find the angle (degrees) whose sine is 1/2. The check takes the sine of your angle and compares to 0.5 — so you must find the angle.

✓ pass

\mathrm{abs}\left(\mathrm{sin}\left(\frac{ang_{you} \cdot \pi}{180}\right) - 0.5\right) < 0.000001

green when sin(your angle) = 1/2

Where next: Trigonometry B turns the circle into waves and handles any triangle with the laws of sines & cosines.