Cos 2x=0.32

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?

Solution:

1. The formula for the volume of a sphere is 
2. r = 6
3. = 

What is an Inverse Function?

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.

Solve them by substitution method: 5b – 3a = 2, (a-1)^2 + (b-1)^2 = 34

Solution:

First, we can write  

Then we can substitute this into the second equation you get

We can simplify this to

If multiply both sides by 25 and expand this out we get

This can be simplified to: Or

Thus the solutions for a are 6 and -4

The solutions for b can be found by the equation (b=4 when a=6, and b=-2 then a=-4)

• Another way to solve this is by looking at the equations graphically. When you plot the two equations, you can see there they intersect.

As you can see, they intersect at two places:

You can plug the answers into both equations to confirm that they are both true using these two values of a and b.

Is this funtion linear? if so, rewrite in y=b+mx. g(w)= 1-12w/3

Solution:

• g(w)=1-12w/3

• g(w)=1 + (-4)w

• In the above example, y=g(w), b=1 and m=-4. If we replace each of the terms we can rewrite the above as y=b+mx

2x-5y=9 and 3x+4y=25. Solve by the method of elimination.

Solution:

1.    3x + 4y=25
2.    2x – 5y=9

We could rewrite these as follows by multiplying the first equation by 5 and the second by 4:

1.    15x + 20y = 125
2.    8x  – 20y = 36

If we add the above two equations, we can eliminate y:

23x = 161

So x =3D 7, and plugging this into the first equation we can figure out
y:  21 + 4y =3D 25,

so y = 1

Solve this problem by using elimination: 2x-3y=61 and 2x+y=-7

Solution:

1. 2x –  =y=61
2. 2x +   y= -7

If you subtract the second equation from the first, you can eliminate x and you are left with -4y = 68.

So y =3D -17, and plugging this into the first equation we can figure out

y:  2x = (-51)=61,

so x = 5

What is the difference b/w relation and function. It would be great if you could give some real life example.

Solution:

A relation is a set of ordered pairs, such as { (0,1) , (55,22), (3,-50) } (http://www.mathwarehouse.com/algebra/relation/math-function.php).  A function is a special kind of relation in which each x value has only one y value.

For example:

Use a half angle identity to find the exact value- sin(75 degrees)

Solution:

The line of a half angle identity is as follows:

If we plug in 75 degrees we get:

12 out of 1000 women at age forty who participate in routine screening have breast cancer. 880 out of 1000 women with breast cancer will get positive mammographies. 90 out of 1000 women without breast cancer will also let positive mammographies. If 100 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?

Solution:

1.    Out of 100 women, we expect 1.2 to actually have breast cancer (12 out of 1000)
2.    However, only 88% (880 out of 1000) of these will be detected, so 1.056 (i.e. 88% of 1.2) will be detected
3.    In addition, of the 98.8 women who don’t have breast cancer 9% (90 out of 1000) will trigger a false positive” (9% of 98.8 = 8.892)
4.    So out of 100 women, 9.948 will have positive mammography readings (1.056+9.892)
5.    Of these 9.948, only 1.056 will actually have breast cancer
6.    10.6% = (1.056/1.056+9.892) This ratio is essentially an application of Bayes Theorem

Please click on http://www.thegodofreason.com/bayesintro.pdf to download a PDF that explains how Bayes Theorem works. Page 3 gives an example that is very similar to your problem: