We can define the basic trigonometric functions using the unit circle. A central angle determines a point P on the unit circle. This angle also determines a right triangle whose hypotenuse is the radius of the unit circle, with length 1.

The other sides of this triangle have lengths and . So, point P has coordinates .

Using the Pythagorean Theorem, we know that . Because and , we have .

Use the tool to change the value of . The sine of is plotted on the grid. You can also choose to plot cosine values of . Click "Done Exploring" when finished.

So the period of both the sine and cosine functions is .

As increases and P moves around the circle, the values of and oscillate between 1 and -1 and eventually repeat.

Look at the graphs of the sine and cosine functions. Notice that the graph of y = sin(x) is symmetric about the origin, so sine is an odd function. Therefore, sin(-x) = -sin(x).

The graph of y = cos(x) is symmetric about the y-axis, so cosine is an even function. Therefore, cos(-x) = cos x.

After the point P has moved one time around the circle, the values of sinx and cosx start to repeat. Therefore, these functions are periodic. The period of any periodic function is the time needed for the function to complete one full cycle.

For any periodic function, the amplitude is half the distance between the maximum and minimum values, if they exist. The maximum and minimum values of sine and cosine are 1 and -1 because these are the maximum and minimum amounts of y and x on the unit circle. So the amplitude of sinx and cosx is 1.

The graph of the sine and cosine functions have exactly the same shape, but one is horizontally shifted. You can get the graph of cosine by shifting the graph of sine units to the left. So, .

Similarly, you can get the graph of sine by shifting the graph of the cosine function units to the right. So, .

We can use the fact that the sine and cosine functions are periodic to evaluate the value of sine and cosine of different angles.

On the unit circle, the endpoint of the angle has coordinates .

So, and .

The angle has the same endpoint on the unit circle as the angle . So, and .

The unit circle is symmetric about the x-axis, the y-axis, and the origin. So, if the coordinates of one point on the circle are known, you can determine the coordinates of three other points. For example, the coordinates of the points Q, R, and S can be found by knowing the coordinates of the point P.

Q is the endpoint of the angle , or . The coordinates of Q are .

So, and .

R is the endpoint of the angle , or . The coordinates of R are . So, and .

Similarly, S is the endpoint of or .

So, and .

Use the tool to change the values of A, B, C, and D. Notice how each affects the graph of y = sinx. You can also choose to view the graph of y = cosx.

How are the amplitude, period, horizontal shift, and vertical shift related to these values? Click "Done Exploring" when finished.

For the periodic functions and , |A| is the amplitude. The period is . Each function is shifted from the original horizontally by C units and vertically by D units. So the midline of the new graph is the line y = D.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education