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## Graphs of Trigonometric Functions

 We have seen that the sine and cosine functions can be constructed geometrically in terms of a unit circle centered at the origin. This applet shows the relationship between the values of the sine, cosine and tangent on the unit circle and their respective graphs. Once the applet is open, click on the exercises button for a full explanation. Can't see the above java applet? Click here to see how to enable Java on your web browser. (This applet was developed by Franz Embacher and Petra Oberhuemer (maths online) and is used here with permission)

The graphs of sin x, cos x and tan x are periodic. A periodic function is one that repeats its values after a period has been added to the independent variable, in this case x. The functions sin x and cos x both have periods equal to 2π. That is, sin x = sin(x + 2π) = sin(x + 4π) = sin(x + 2kπ) for any integer k and cosx = cos(x + 2π) = cos(x + 4π) = cos(x + 2kπ) for any integer k. Their graphs are each shown below. Notice that the distance between successive peaks (or successive troughs) in the graphs of sin x and cos x is equal to the period 2π. This distance is known in physics as the wavelength. The amplitude of each of the functions sin x and cos x is 1. This refers to the distance from the peak (or trough) and the baseline (the horizontal line located halfway between the peak and trough, in this case the x-axis) or half the vertical distance from the peak to trough.    The graph of tan x is periodic but the period is π. So tan x = tan(x + kπ) for any integer k.  Exercise 7.

 How many solutions are there to the equation in the interval [-2π , 2π]

For the function y = a*sin(b*x + c) + d, the amplitude is given by the value of the parameter a. The period of the function is given by 2π/b. The parameter c shifts the function horizontally to the left by the given amount and d shifts the function vertically upwards by the given amount. Use the applet below to see visually how this works. Move the sliders to adjust the values of a, b, c and d in y = a*sin(b*x+c)+d.

For the function y = a*cos(b*x + c) + d, the parameters a, b, c and d have the same effect on the cosine function as they did on the sine. Use the applet below to see visually how this works. Move the sliders to adjust the values of a, b, c and d in y = a*cos(b*x+c)+d.

### Exercise Applets

 Match the functions of the form   a sin(b + c x)  or   a cos(b + c x)  with their graphs. The applet is started by clicking on the red button and will open in its own window.

 Match the graphs of the functions of the form a sin(b + c x)  or   a cos(b + c x)  with their functions. The applet is started by clicking on the red button and will open in its own window.