Trigonometric Functions of any angle in terms of Cartesian coordinates

Setting up standard position. Place the vertex of an angle $\theta$ at the origin $O$ with its initial side along the positive $x$-axis; the angle is measured (anticlockwise for positive $\theta$, clockwise for negative $\theta$) to the terminal side. Let $P(x,y)$ be any point other than the origin on the terminal side, and let $r=OP=\sqrt{x^2+y^2}>0$ be its distance from the origin.

Definition (trigonometric functions of any angle).

and from these, $\tan\theta=\dfrac{y}{x}\ (x\ne0)$, $\cot\theta=\dfrac{x}{y}\ (y\ne0)$, $\operatorname{cosec}\theta=\dfrac{r}{y}\ (y\ne0)$, $\sec\theta=\dfrac{r}{x}\ (x\ne0)$.

A few immediate consequences:

• Since $|x|\le r$ and $|y|\le r$ always (because $r=\sqrt{x^2+y^2}\ge |x|,|y|$), we get $|\sin\theta|\le1$ and $|\cos\theta|\le1$ for every angle $\theta$.
• For an acute angle these definitions agree exactly with the familiar right-triangle ratios — $r$ plays the role of the hypotenuse, $y$ the opposite side, $x$ the adjacent side.
• Because $x,y$ can each be positive or negative depending on which quadrant $P$ lies in (while $r>0$ always), the six functions pick up positive or negative signs quadrant by quadrant — developed below.
• The value of each function depends only on $\theta$, never on which particular point $P$ was chosen on the terminal side: any two points on the same ray give similar right triangles, so every ratio like $y/r$ comes out identical.

Quadrantal angles. An angle in standard position whose terminal side falls exactly on one of the axes (so $\theta=0^\circ,90^\circ,180^\circ,270^\circ,360^\circ,\ldots$) is a quadrantal angle. Take $P$ on the unit circle $x^2+y^2=1$ (so $r=1$); then directly from the definition, $\cos\theta=x$ and $\sin\theta=y$ — the coordinates of $P$ are $(\cos\theta,\sin\theta)$.

Reading the zeros of sine/cosine straight off the table gives two very useful generalizations, valid for every integer $n$:

Since $\tan\theta=\sin\theta/\cos\theta$, it follows that $\tan\theta$ is undefined exactly at the zeros of $\cos\theta$, i.e. at $\theta=(2n+1)\pi/2$.

Signs of the trigonometric functions (the ASTC rule). Take $P(x,y)$ on the unit circle, so $\cos\theta=x$, $\sin\theta=y$, $\tan\theta=y/x$. The sign of each function is decided purely by the signs of $x$ and $y$ in the quadrant containing $P$:

A handy mnemonic for which functions are positive, quadrant by quadrant (I, II, III, IV): 'All Students Take Chocolate' — All, Sine (and cosecant), Tangent (and cotangent), Cosine (and secant).

Worked illustrations of the method. …