The sine function in one of the basic three trigometric functions: sine, cosine, and tangent. These trig functions represent ratios of the sides of a right triangle. On the triangle below, notice the sides. The longest ,which is across from the right angle, is the hypotenuse. The other two legs are labeled adjacent and opposite according to their relationship to the chosen angle.

The sine function is the ratio of the opposite side to the hypotenuse. The cosine function is the ratio of adjacent side to the hypotenuse. The tangent function is the ratio of the opposite to the adjacent. An easy acronym to help remember this is SOH CAH TOA. Since the hypotenuse is the longest side, the sine ratio should always be less than one.

Now let's look at the sin function on the unit circle. The unit circle has a raduis of one unit with center at (0,0). We can graph this circle by the equation .This circle is divided into sections using degrees or radians. A complete circle is 360 degrees or 2 radians. Notice several key points in the first quadrant, which will only be from 0 to /2

Now look at these points as they are graphed.

This creates the sine curve which is continuous. However, the curve completes a cycle between 0 and 2 . The range of the sine curve is -1 <= y<=1. Zoming out shows the graph over a larger domain. Remember this sine graph will not have values in the range greater than one but will continue repeating the curve.

The basic sine graph can be transformed several ways. First, the amplitude can be increased by multiplying the function by a constant. This is the a in y = a sin x. For example, the graph y=2sin x changes the amplitude to 2.

This will happen with any value plugges into y=a sin x. When 5 is given to value of 5, the amplitude will increase to 5. Now let's check out the negative numbers. If y = -sin x the graph will be reflected over the x-axis. Instead of increasing from 0 to /2 the graph will now decrease

Now, let's consider the next variable, b, in y = sin (bx).

Notice the graph now. The amplitude is still one, but there are now two cycles of the graph between O and 2 pi. This is true for any value of b. The number previous to x is the number of cycles the graph will have between O and 2 pi. For instance, if b is 0.5, the graph will have half the cycle between O and 2 pi. Notice the graph below.

Finally, consider the changes c will have on the sine graph of

y=sin (x +). The value of c changes the phase shift. The previous equation will be translated units to the right. Notice the graph below.

What was once at 0 has now been moved over to . Any value of c will translate the graph in the same number of units.

In Summary

a, b, and c have the following effects on the equation y = a sin (bx +c)

a --amplitude of graph

b -- number of cyles between 0 and 2 pi

c -- the phase shift

The sine graph always follows these basic rules. This is helpful to graph the equations, especially graphing without the calculator. A solid understanding of the graphs will help in working with trigometric equations.

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