Certainly, readers will have experienced different definitions for these functions. It can be very reassuring to see these functions as signed lengths instead of just ratios of each other. It has the added benefit of containing geometric explanations for many common identities --- Construct or observe the following diagram: - point $P$ is on a Unit Circle $O$ - point $X$ is a projection of $P$ onto the $x$-axis - point $Y$ is a projection of $P$ onto the $Y$-axis - the tangent to $O$ at point $P$ has $x$ and $y$ intercepts at $A$ and $B$ respectively - angle $\theta$ is the angle from the positive $x-axis$ to the radius $\overline{OP}$ ![[def-of-trig-funcs.svg|-dmo-noinv]] Define the following functions: - [[Cosine Function]] $\cos\theta=m\overline{YP}$ with a sign corresponding with the direction of $\overrightarrow{YP}$ - [[Sine Function]] $\sin\theta=m\overline{XP}$ with a sign corresponding with the direction of $\overrightarrow{XP}$ - [[Secant Function]] $\sec\theta=m\overline{OA}$ with a sign corresponding with the direction of $\overrightarrow{OA}$ - [[Cosecant Function]] $\csc\theta=m\overline{OB}$ with a sign corresponding with the direction of $\overrightarrow{XB}$ - [[Cotangent Function]] $\cot\theta=m\overline{BP}$ with the sign opposite of its slope - [[Tangent Function]] $\tan\theta=m\overline{AP}$ with the sign opposite of its slope --- $\triangle OXP$ (or the congruent $\triangle OYP$), $\triangle PXA$, and $\triangle BPO$ can be used to explain the [[Pythagorean Identities]]