Every point on the unit circle is \((\cos\theta,\;\sin\theta)\). The x-coordinate is cosine, the y-coordinate is sine.
Example: at \(\frac{\pi}{4}\) (45°), the point is \(\left(\frac{1}{\sqrt{2}},\;\frac{1}{\sqrt{2}}\right)\), so \(\cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\) and \(\sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}\).
| Degrees | Radians | \(\cos\theta\) | \(\sin\theta\) |
|---|---|---|---|
| \(0°\) | \(0\) | \(1\) | \(0\) |
| \(30°\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) |
| \(45°\) | \(\dfrac{\pi}{4}\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{\sqrt{2}}\) |
| \(60°\) | \(\dfrac{\pi}{3}\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) |
| \(90°\) | \(\dfrac{\pi}{2}\) | \(0\) | \(1\) |
| \(120°\) | \(\dfrac{2\pi}{3}\) | \(-\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) |
| \(135°\) | \(\dfrac{3\pi}{4}\) | \(-\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{\sqrt{2}}\) |
| \(150°\) | \(\dfrac{5\pi}{6}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) |
| \(180°\) | \(\pi\) | \(-1\) | \(0\) |
| \(210°\) | \(\dfrac{7\pi}{6}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(-\dfrac{1}{2}\) |
| \(225°\) | \(\dfrac{5\pi}{4}\) | \(-\dfrac{1}{\sqrt{2}}\) | \(-\dfrac{1}{\sqrt{2}}\) |
| \(240°\) | \(\dfrac{4\pi}{3}\) | \(-\dfrac{1}{2}\) | \(-\dfrac{\sqrt{3}}{2}\) |
| \(270°\) | \(\dfrac{3\pi}{2}\) | \(0\) | \(-1\) |
| \(300°\) | \(\dfrac{5\pi}{3}\) | \(\dfrac{1}{2}\) | \(-\dfrac{\sqrt{3}}{2}\) |
| \(315°\) | \(\dfrac{7\pi}{4}\) | \(\dfrac{1}{\sqrt{2}}\) | \(-\dfrac{1}{\sqrt{2}}\) |
| \(330°\) | \(\dfrac{11\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(-\dfrac{1}{2}\) |
| \(360°\) | \(2\pi\) | \(1\) | \(0\) |
For \(0°, 30°, 45°, 60°, 90°\) — cosine is:
$$\frac{\sqrt{4}}{2},\quad \frac{\sqrt{3}}{2},\quad \frac{\sqrt{2}}{2},\quad \frac{\sqrt{1}}{2},\quad \frac{\sqrt{0}}{2}$$
Just count down: 4, 3, 2, 1, 0 under the root. And \(\frac{\sqrt{4}}{2} = 1\), \(\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}\), \(\frac{\sqrt{0}}{2} = 0\).
Sine is the same sequence counting up: \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\)
Other quadrants: same values, just flip signs. Cosine is negative on the left half (Q2, Q3). Sine is negative on the bottom half (Q3, Q4).
\(\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}} \approx 0.707\) — the equal superposition value. Hadamard gate lives here.
\(\cos 0 = 1,\;\sin 0 = 0\) — the \(|0\rangle\) state (north pole of Bloch sphere).
\(\cos\frac{\pi}{2} = 0,\;\sin\frac{\pi}{2} = 1\) — but the Bloch formula uses \(\theta/2\), so \(\theta = \pi\) puts you at the south pole \((|1\rangle)\).