The sine and cosine functions are functions of the central angle of the unit circle. As the measurement of the angle changes, the value of sine changes and the value of cosine changes.

As the angle rotates counterclockwise around the origin, the value of sine sometimes increases and sometimes decreases. We can compare the changes in \(\theta \) with the changes in the value of \(\sin (\theta )\) in the form of a rate.

We can visualize this rate of change as the slope of the tangent lines to the graph of \(\sin (\theta )\).

If we collect these slopes of the tangent lines and plot them on the Cartesian plane, they form a pattern.

The slopes of the tangent line create a function called the instantaneous rate of change (i.e. the derivative) of \(\sin (\theta )\).

It turns out this derivative of \(\sin (\theta )\) is \(\cos (\theta )\).

Learning Outcomes

In this section, students will

• compare rates of change of sine and cosine.