Math Homework. Do It Faster, Learn It Better.

Cosine Function

The cosine function is a periodic function which is very important in trigonometry.

The simplest way to understand the cosine function is to use the unit circle. For a given angle measure θ , draw a unit circle on the coordinate plane and draw the angle centered at the origin, with one side as the positive  x -axis. The x -coordinate of the point where the other side of the angle intersects the circle is cos ( θ ) , and the  y -coordinate is sin ( θ ) .

There are a few cosine values that should be memorized, based on  30 ° 60 ° 90 ° triangles  and  45 ° 45 ° 90 ° triangles .

Once you know these values, you can derive many other values for the cosine function. Remember that cos\theta; is positive in quadrants I and I V and negative in quadrants I I and I I I .

You can plot these points on a coordinate plane to show part of the cosine function, the part between 0 and 2 π .

For values of θ less than 0 or greater than 2 π you can find the value of cos ( θ ) using the reference angle .

The graph of the function over a wider interval is shown below.

Note that the of the function is the whole real line, while the range is 1 y 1 .

The period of f ( x ) = cos ( x ) is 2 π . That is, the shape of the curve repeats every 2 π -unit interval on the x -axis.

The amplitude of f ( x ) = cos ( x ) is 1 , that is, the height of the wave.

The modified function y = a cos ( b x ) has amplitude a and period 2 π / b .