Trigonometric Functions of real numbers

Why extend to real numbers? Calculus, physics and chemistry frequently need $\sin$ and $\cos$ of a plain real number $t$ — not an angle measured in degrees — because these functions are used to model things like waves and oscillations that are described by a real-valued time or position variable. Section 3.4.1 only defined trigonometric functions of an angle; this subsection extends the definition to any real number, by matching each real number to a point on the unit circle.

The wrapping function. Take the unit circle centred at the origin, and fix the point $A(1,0)$ — this corresponds to the real number $0$. Draw the tangent line to the circle at $A$ (a vertical line), and think of it as a real number line with $A$ at its zero. For a real number $t$, mark off an arc of length $|t|$ along the circle starting at $A$: anticlockwise if $t>0$, clockwise if $t<0$. This lands at a point $B(x,y)$ on the circle. If $\theta$ is the angle subtended at the centre by this arc $AB$, then, because the circle has radius $1$, arc length $=$ radius $\times$ angle gives arc length $t=\theta$ — the arc length is the radian measure of the angle. This assignment $t\mapsto B$ is called the wrapping function $w(t)$ (like wrapping a number line around the circle).

Definition. For a real number $t$, define

where $\theta$ (in radians) is the angle produced by wrapping $t$ as above. Equivalently, since $B(x,y)=B(\cos t,\sin t)$ lies on the unit circle, $\sin t=y$ and $\cos t=x$ directly — the $y$- and $x$-coordinates of the wrapped point. The remaining four functions of a real number $t$ are then defined from $\sin t$ and $\cos t$ exactly as before (quotient and reciprocal identities).

A few remarks:

• Because $B(\cos t,\sin t)$ is always a point on the unit circle, $-1\le\cos t\le1$ and $-1\le\sin t\le1$ for every real number $t$, exactly as for angles. …