Sine, cosine, and tangent on the unit circle

Put an angle $\theta$ in standard position: vertex at the origin, initial ray along the positive x-axis, positive angles sweeping counterclockwise. Its terminal ray crosses the unit circle (radius 1, centered at the origin) at exactly one point, and that point defines everything:

1. cos θ is the x-coordinate of the terminal point: how far right or left it sits.
2. sin θ is the y-coordinate: how far up or down it sits.

The order matters and students reverse it constantly. A reliable anchor: the terminal point IS $(\cos\theta, \sin\theta)$, and points are always written $(x, y)$, so cosine rides first. At $\theta = 0$ the point is $(1, 0)$: cosine starts at 1, sine starts at 0. On a circle of radius r instead of 1, the point scales to $(r\cos\theta, r\sin\theta)$; the cosine and sine themselves are the coordinates divided by r.

A radian is the natural angle unit of the circle itself: the measure of $\theta$ in radians equals the arc length it cuts off on the unit circle. In general $\theta = s/r$, arc length over radius. One full trip around the unit circle is the whole circumference, $2\pi$, which is why $2\pi$ radians = 360° and $\pi$ radians = 180°.

Converting is one proportion: multiply degrees by $\pi/180$ to get radians, multiply radians by $180/\pi$ to get degrees. The landmarks are worth knowing on sight: 30° = $\pi/6$, 45° = $\pi/4$, 60° = $\pi/3$, 90° = $\pi/2$, 180° = $\pi$, 270° = $3\pi/2$.

The costly error is treating a bare number like 2 as if it were 2°. The number 2 with no degree mark is 2 radians, nearly a third of the way around the circle (about 114.6°). The AP exam runs in radians; so does every formula in this unit.

Define $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. On the picture this is $y/x$ for the terminal point, which is precisely the slope of the terminal ray. A ray climbing steeply has a large tangent; a shallow ray has a small one; a ray sloping down has a negative one.

The slope reading explains tangent's quirks before you ever graph it. Where the terminal ray is vertical ($\theta = \pi/2$ or $3\pi/2$), the ray has no slope, $\cos\theta = 0$, and $\tan\theta$ is undefined. And because a ray and its opposite ray have the same slope, tangent repeats every half turn, a fact that becomes its $\pi$ period in Topic 3.8.

Adding or subtracting a full turn, $2\pi$ radians (360°), leaves the terminal ray exactly where it was. Angles that share a terminal ray are coterminal, and they share all three trig values: $\theta$, $\theta + 2\pi$, $\theta - 2\pi$, and so on are trig-identical. Note it is the full $2\pi$, not $\pi$: adding half a turn lands on the opposite ray, which flips the signs of sine and cosine.

Finally, the mode trap. A calculator in the wrong angle mode answers a different question than the one you asked: sin(30) in radian mode evaluates the sine of 30 radians, about $-0.988$, nothing like the $0.5$ of $\sin(30^\circ)$. The AP exam requires radian mode. Check the mode before you trust a single decimal.