Sine / Cosine For Point ,\(P\), Off Unit Circle

It was seen that for a point on the unit circle with coordinates \((x,y)\), \(x = \cos \theta\) and \(y = \sin \theta\). The question becomes, what about for a point, \(P\), which is off the unit circle.

The following diagram will help explore this.

The point \(P\) now has coordinates \((x,y)\). From the diagram it can be seen that two perpendiculars are dropped to the x-axis, the first being from where the line \(OP\) intersects the unit circle (at point \(B\)) and from the point \(P\) itself. What results are two similiar triangles which have the property that corresponding sides are in the same proportion. The two similiar trianges are \(\Delta AOB\) and \(\Delta COP\). From this then:

\[ \begin{align} \frac{|OC|}{|OA|} &= \frac{r}{1} \\ \frac{|OC|}{|OA|} &= r \\ \end{align} \]

(Note: Using notation \(|OC|\) to refer to the length of the line between points \(O\) and \(C\); it is sometimes also written as \(\overline{OC}\))

The coordinate of \(C\) is \((x,0)\) so \(|OC| = x\). Since the point \(A\) is from the perpendicular dropped from the point on the unit circle \(\cos \theta = |OA|\). Making these substitutions yields:

\[ \begin{align} \frac{|OC|}{|OA|} &= r \\ \frac{x}{\cos \theta} &= r \end{align} \]

Algebraic manipulation produces

\[ \cos \theta = \frac{x}{r} \]

and

\[ x = r \cos \theta \]

Similiar logic can be applied for the \(\sin \theta\).

\[ \begin{align} \frac{|CP|}{|AB|} &= \frac{r}{1} \\ \frac{|CP|}{|AB|} &= r \\ \end{align} \]

Recognizing that \(\sin \theta = |AB|\) and \(|CP| = y\), the following substitutions can be made:

\[ \begin{align} \frac{|CP|}{|AB|} &= r \\ \frac{y}{\sin \theta} &= r \end{align} \]

Algebraic manipulation produces

\[ \sin \theta = \frac{y}{r} \]

and

\[ y = r \sin \theta \]

Hence, for any point, \(P\), with coordinates \((x,y)\) of distance \(r\) from the origin, \(x = r \cos \theta\) and \(y = r \sin \theta\).

(Note that these are the general equations for, when applied to the unit circle, \(r = 1\) so they reduce to simply \(x = \cos \theta\) and \(y = \sin \theta\).)

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