Trigonometric Functions via the Unit Circle

The definition of $\sin\theta$ and $\cos\theta$ as ratios of sides in a right

triangle only makes sense for an acute angle $\theta$ (strictly between $0^\circ$ and

$90^\circ$), since a right triangle cannot have an angle of $120^\circ$ or $-40^\circ$. To

define the trigonometric functions of any real number $\theta$ -- positive, negative, or

larger than $360^\circ$ -- the unit circle is used instead.

Setting up the unit circle. Let $\theta$ be any real number, taken as an angle in standard

position (vertex at the origin $O$, initial side along the positive x-axis, Section 1). Let the

terminal side of $\theta$ meet the circle of radius $1$ centred at the origin -- the unit circle, $x^2+y^2=1$ -- at the point $P(x,y)$.

Definition. The cosine and sine of $\theta$ are defined as the coordinates of this point:

The remaining four trigonometric functions are then defined as ratios built from $x$ and $y$,

exactly as their names suggest they should relate to $\sin\theta$ and $\cos\theta$:

This definition agrees with the right-triangle definition. For an acute angle $\theta$,

drop a perpendicular from $P(x,y)$ to the x-axis, forming a right triangle with the origin and

the foot of the perpendicular. Its hypotenuse is $OP=1$ (the radius), its side adjacent to

$\theta$ has length $x$, and its side opposite $\theta$ has length $y$. Then

$\text{adjacent}/\text{hypotenuse}=x/1=x=\cos\theta$ and

$\text{opposite}/\text{hypotenuse}=y/1=y=\sin\theta$ -- exactly the familiar right-triangle

ratios. The unit-circle definition is therefore not a different idea but a genuine

extension of the same one to every real $\theta$, since $\theta$ no longer needs to be a

angle of an actual triangle at all.

Why this makes every function periodic. Because $\theta$ and $\theta+2\pi$ are coterminal

(Section 1), they place the terminal side -- and hence the point $P$ -- in exactly the same

position. So $\cos(\theta+2\pi)=\cos\theta$ and $\sin(\theta+2\pi)=\sin\theta$ for every real

$\theta$: sine and cosine repeat every $2\pi$, which is precisely the period recorded in the

graphs of Section 5. …