The Unit Circle

Triangle in unit circle

Drag the red point!

Since the trigonometric ratios do not depend on the size of the triangle, you can always use a right-angled triangle where the hypotenuse has length one. If you place such a triangle in a Cartesian system, as in the applet above, the vertex \(A\) will lie on a circle with radius one. A circle having the radius one is called a unit circle. When the hypotenuse is one, the values of sine and cosine are:

\[\sin \alpha = \frac{\text{opp}}{\text{hyp}}=\text{opp} \hspace{1cm} \cos \alpha = \frac{\text{adj}}{\text{hyp}}=\text{adj}\]

As can be seen in the applet, it is also true that \(\sin \alpha = y(A)\) and \(\cos \alpha = x(A)\) where \(x(A)\) and \(y(A)\) are the \(x\)- and \(y\)-coordinate respectively of \(A\). Using the coordinates of \(A\), instead of ratios of sides in a right-angled triangle, we can extend the definition of sine and cosine so they are defined for all angles.

In GeoGebra, all angles are shown as positive and less than \(360^\circ\). However, if an angle is extended from the positive \(x\)-axis in a clockwise order, it is a negative angle. Furthermore, angles can be larger than \(360^\circ\). Using the unit circle definition of sine and cosine, these functions are defined for all angles. Note that the triangle definition is still valid for an angle \(\alpha\) if \(0\lt \alpha \lt 90^\circ\).

The graphs of sine and cosine

Drag the red point to draw the graphs!

If you enter sin(x) or cos(x) in the input bar in GeoGebra, you will see the graphs of sine or cosine for all values of \(x\). The parts of the graphs that are plotted in the applet above, are repeated infinitely many times. The functions are periodic and the period is \(360^\circ\).

If \(f(x)\) is a periodic function with period \(P\), then \(f(x)=f(x+P)\) for all \(x\).

Plotting trigonometric functions using degrees in GeoGebra

You may notice that the scale on the \(x\)-axis does not correspond to the angle. The reason for this is that GeoGebra uses the unit radians instead of degrees for angles. If you want GeoGebra to use degrees, you must enter the degree-symbol, press Ctrl+o to write the degree-symbol. The code entered should be sin(x°) or cos(x°).

Image

Rescale the \(x\)-axis by pressing Shift and then drag the \(x\)-axis. In the properties window for the graphics view, you can change the distance between labels and add the degree symbol as a unit.

Extending the definition of tangent and cotangent

Drag the red point!

Using the triangle definition, tangent and cotangent are defined as

\[\tan \alpha = \frac{\text{opp}}{\text{adj}} \hspace{1cm} \cot \alpha = \frac{\text{adj}}{\text{opp}}\]

Exercise 1

Using the construction in the applet above, show that \(\tan \alpha = y(P)\) for all \(0\lt \alpha \lt 90^\circ \).

Exercise 2

Using the construction in the applet above, show that \(\cot \alpha = x(Q)\) for all \(0\lt \alpha \lt 90^\circ \).

The graphs of tangent and cotangent

Drag the red point to draw the graphs!

Tangent and cotangent are periodic functions with the period \(180^\circ \).

\(\tan \alpha\) is not defined when \(\alpha = 90^\circ +n\cdot 180^\circ, n \in \mathbb{Z} \).

\(\cot \alpha \) is not defined when \(\alpha = n\cdot 180^\circ, n \in \mathbb{Z} \).

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se