# Cos 2x=0.32

Monday, September 6th, 2010

### Solution:

First let’s apply the reverse cosine function to both sides of the equation:

cos-1(cos(2x)) = cos-1 (0.32) = cos-1 ()

Now if you take the inverse cosine the  cosine of 2x, you get 2x…so:

2x = cos-1 ()

2x 71.34°

x 35.67°  (which is approximately the same as 0.622 radians)

This looks like it might be the final answer, but actually it’s only one of the many correct answers. One way to see this is by graphing y=cos(2x) and seeing where it intersects the line y=0.32. If you do this, you will see that the intersections occur at multiple points:

As you can see, the intersections occur at x 0.622 radians, 2.519 radians, 3.764 radians, 5.660 radians, etc. The general form of the solution set is as follows:

# What is the Volume of a Sphere the Radius of Six?

Saturday, September 4th, 2010

### Solution:

1. The formula for the volume of a sphere is
2. r = 6

# What is an Inverse Function?

Saturday, September 4th, 2010

If you have a function called ƒ, let’s call its inverse ƒ–1 (i.e. ƒ–1 is the inverse function of ƒ).

By definition, the property of ƒ–1 is that if ƒ(a)=b, then ƒ–1(b)=a. Wikipedia (http://en.wikipedia.org/wiki/Inverse_function) has a good example:

A function ƒ and its inverse ƒ–1. Because ƒ maps a to 3, the inverse ƒ–1 maps 3 back to a. In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip from A to B to A (or from B to A to B) returns each element of the initial set to itself.