Graphs of Sine and Cosine

Graphing the trigonometric functions - Sine and Cosine

The sine of an angle is defined to be the y-coordinate of the point on the unit circle.

The cosine of an angle is defined to be the x-coordinate of the point on the unit circle.

The unit circle produces wave-like curves for $y=\sin\theta$ and $y=\cos\theta$.

*Notice when the functions have positive and negative values. Match them up with the corresponding quadrants and check they behave in the same way.

Properties of Circular Graphs

wave functions, periodic graphs, cyclic graphs, oscillating graphs…

1. They demonstrate repetition. One cycle (wavelength), for these graphs, is 360°.

2. The possible value of y for sine (and cosine) waves is -1 and 1. This means the range: $-1\le y\le1$ for both $y=\sin\theta$ and $y=\cos\theta$

3. For the cycles to repeat, there has to be regions on the graph where y is increasing (as x increases) and other regions where y is decreasing (as x increases).

Graphing the trigonometric functions - Tangent

The tangent of an angle is defined as the ratio $\left(\frac{\sin\theta}{\cos\theta}\right)$.

We also know that the gradient$m$ of a straight line is given by $\frac{rise}{run}$.

This means $\tan\theta$ is just the gradient of the line $OP$ raised $\theta$ from the positive x-axis.