Question

List all of the values of the sine function that you know. Remember that values of sin(x) repeat every 2π radians, so your answer should include infinitely many values.

Answer 1

The graph of y = sin θ

Using the values sin 30° = = 0.5 and sin 60° = ≈ 0.87, we can draw up the following table of values and then plot them.

θ°		0		30		60		90		120		150		180		210		240		270		300		330		360
sin θ		0		0.5		0.87		1		0.87		0.5		0		−0.5		−0.87		−1		−0.87		−0.5		0

More points can be used to show that the shape of the graph is as shown below.

Electrical engineers and physicists call this a sinewave.

You can see many interesting symmetries in the graph of .

The model employing the unit circle helps to elucidate these. The sine of the angle θ is represented by the y-value of the point P on the unit circle. Thus, since sin θ = sin (180 − θ), we mark the two equal intervals in the graph.

Hence, between 0° and 180°, the graph is symmetric about θ = 90°.

Similarly, between 180° and 360°, the graph is symmetric about θ = 270°.

Finally, the graph possesses a rotational symmetry about θ = 180° as the following diagrams demonstrate.

All these observations are summarised by the diagram:

This symmetry diagram illustrates the related angle, the quadrant sign rules and the symmetries discussed above.

Extending the graph

We noted above that the values of sine repeat as we move through an angle of 360°, that is, sin (360° + θ) = sin θ . We say that the function y = sin θ is periodicwith period 360°. Thus, the graph may be drawn for angles greater than 360° and less than 0°, to produce the full (or extended) graph of y = sin θ.

Note that the extended sine graph has even more symmetries. There is a translation (by 360°) symmetry, a reflection symmetry about any odd multiple of 90° and a rotational symmetry about vertical lines through any even multiple of 90°.