The Definition of Everything
On the unit circle (where radius \(r=1\)), the trig ratios become incredibly simple. For any point \((x, y)\) on the circle at an angle \(\theta\):
- \(\cos\theta = x\)
- \(\sin\theta = y\)
- \(\tan\theta = y/x\)
The coordinate of every point is just \((\cos\theta, \sin\theta)\).
Signs in Quadrants (ASTC)
How to remember where sin/cos/tan are positive:
- Q I (All): All are positive.
- Q II (Sin): Sine is positive (y is up), Cos is negative.
- Q III (Tan): Both x and y are negative, so their ratio (tan) is positive.
- Q IV (Cos): Cosine is positive (x is right), Sin is negative.
Worked Examples
Example 1: Evaluating at Quadrant Angles
Find \(\sin(180^\circ)\) and \(\cos(180^\circ)\).
- At 180°, the point is on the negative x-axis: \((-1, 0)\).
- \(\cos(180^\circ) = -1\).
- \(\sin(180^\circ) = 0\).
Example 2: Using Reference Angles
Find \(\sin(210^\circ)\).
- 210 is in Q III. The distance from the x-axis (reference angle) is \(210 - 180 = 30^\circ\).
- \(\sin(30^\circ) = 0.5\).
- Since we are in Q III, sine must be negative.
- Result: -0.5
The Bridge to Quantum Mechanics
In Quantum Mechanics, we use the Unit Circle to represent the Probability Amplitude. The "length" of the vector is always 1 (normalization), but its direction (the angle \(\theta\)) determines how it interacts with other waves. This is the origin of the "Wavefunction" being a complex number: we use the x-axis for the "Real" part and the y-axis for the "Imaginary" part. Every particle in the universe is essentially a point moving around this circle.