In other modules, the unit circle was introduced and used to determine the exact trigonometric ratios for multiples of \(\dfrac{\pi}{6} , \dfrac{\pi}{4} , \dfrac{\pi}{3}\), and \(\dfrac{\pi}{2}\) radians and multiples of \(30^{\circ}, 45^{\circ}, 60^{\circ}\), and \(90^{\circ}\).

We discovered that any point, \(P\), on the unit circle and also on the terminal arm of a standard position angle, \(\theta\), can be written \(P\,(\cos(\theta), \sin(\theta))\).

We will use this background to help us draw the graphs for the three primary trigonometric functions.

New terminology will be introduced as we examine properties associated with these graphs.

Primary Trigonometric Functions:

\(y = \sin(\theta)\), \(y = \cos(\theta)\), \(y = \tan(\theta)\)

Throughout this module, and in fact for the rest of the unit, you will also see \(y = \sin(x)\), \(y = \cos(x)\), and \(y = \tan(x)\) where \(x\) is used instead of \(\theta\).

We know the coordinates of points on the unit circle corresponding to standard position angles whose reference angles are \({\color{NavyBlue}\dfrac{\pi}{6}}\) , \({\color{BrickRed}\dfrac{\pi}{3}}\) , \({\color{Mulberry}\dfrac{\pi}{4}}\), and \({\color{Violet}\dfrac{\pi}{2}}\) radians. We also know that the \(y\)-coordinate of any point on the terminal arm of a standard position angle \(x\) and the unit circle is \(\sin(x)\).

Construct a table of values relating \(x\) and \(\sin(x)\) for \(0 \leq x \leq 2 \pi\). Use increments of \(\dfrac{\pi}{12}\) radians (this corresponds to \(15^{\circ}\) intervals). Fill in the exact values for \(\sin (x)\) by referring to the appropriate unit circle diagram. We have not determined the exact values for \(\sin (x)\) for which the reference angle is either \(\dfrac{\pi}{12}\) or \(\dfrac{5\pi}{12}\) so we will calculate approximate values later. In the table, these values are shown as “?”. Some have been filled in for you.

Construct a table of values relating \(x\) and \(\sin(x)\) for \(0 \leq x \leq 2 \pi\). Use increments of \(\dfrac{\pi}{12}\) radians (this corresponds to \(15^{\circ}\) intervals). Fill in the exact values for \(\sin (x)\) by referring to the appropriate unit circle diagram. We have not determined the exact values for \(\sin (x)\) for which the reference angle is either \(\dfrac{\pi}{12}\) or \(\dfrac{5\pi}{12}\) so we will calculate approximate values later. In the table, these values are shown as “?”. Some have been filled in for you.

From the table of values, we can plot points to determine the shape of the function \(y = \sin(x)\).

The exact values of \(\sin(x)\) have been converted to approximate values in the following table, correct to two decimals, where necessary.

A calculator was used to determine the approximate values for multiples of \(\dfrac{\pi}{12}\).

These were marked with a “?” mark in the previous chart.

Plot the points on a graph. The \(x\)-axis uses intervals of \(\dfrac{\pi}{12}\) and the \(y\)-axis uses intervals of \(0.2\).

Connect the points on the graph with a smooth curve.

What happens if we continue our table from \(2 \pi\) to \(4 \pi\)?

It is relatively easy to see that the values in the table will repeat since we are passing through the same points on the unit circle each time we complete a full rotation.

Another way to discuss this is through the use of coterminal angles.

For example, \(\dfrac{25 \pi}{12}\) is coterminal with \(\dfrac{\pi}{12}\).

The terminal arm will intersect the unit circle at the same point for both \(\dfrac{25 \pi}{12}\) and \(\dfrac{\pi}{12}\). It follows that \(\sin \left( \dfrac{25 \pi}{12} \right) = \sin \left( \dfrac{\pi}{12} \right)\).