Unit Circle Chart & Calculator

The unit circle is the circle $x^2 + y^2 = 1$ centred at the origin. Measure an angle $\theta$ counter-clockwise from the positive $x$-axis and the point where the terminal ray meets the circle is exactly

$(x, y) = (\cos\theta,\ \sin\theta), \qquad \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}$

So cosine is the $x$-coordinate, sine is the $y$-coordinate, and tangent is the slope of the ray. Because the radius is $1$, the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ is just the circle's own equation.

Signs by quadrant (ASTC): I all positive; II sine only; III tangent only; IV cosine only. Angles repeat every $2\pi$ for $\sin$ and $\cos$, and every $\pi$ for $\tan$. Reciprocals follow directly: $\sec\theta = 1/x$, $\csc\theta = 1/y$.

The $\frac{\sqrt{n}}{2}$ pattern. For $0°, 30°, 45°, 60°, 90°$ the sine values are

$\frac{\sqrt{0}}{2},\ \frac{\sqrt{1}}{2},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{3}}{2},\ \frac{\sqrt{4}}{2}$

and the cosine values are the same list read backwards. That single pattern gives you the whole first quadrant.

Extend by reference angle. Every other standard angle reuses a first-quadrant value: take the acute angle to the $x$-axis, look it up, then attach the ASTC sign for the quadrant.

Radians without conversion: denominators $6, 4, 3$ correspond to $30°, 45°, 60°$; the numerator counts how far round you are.

Common mistakes

• Swapping $\sin$ and $\cos$ — remember $x$ comes first, and $x$ is cosine.
• Measuring the reference angle to the $y$-axis instead of the $x$-axis.
• Writing $\tan 90° = \infty$: it is undefined, because $\cos 90° = 0$.
• Leaving a calculator in degree mode while the question is in radians.