geno3141

The Law of Cosines is:  c²  =  a² + b² - 2a·b·cos(C)

For an example:

If  a = 3   b = 5   c = 7   

--->   7²  =  3² + 5² - 2·3·5·cos(C)

--->   49  =  9 + 25 - 30cos(C)

--->   49  =  34 - 30cos(C)

--->   15  =  -30cos(C)

--->   15/-30  =  cos(C)

--->   -0.5  =  cos(C)

--->   C  =  cos^-1(-0.5)

--->   C  =  120°

Since  cos(2θ)  =  2cos²(θ) - 1,

cos(2θ) + 2cos(θ) + 1  =  0

--->   2cos²(θ) - 1 + 2cos(θ) + 1  =  0

2cos²(θ) + 2cos(θ)  =  0

cos²(θ) + cos(θ)  =  0

cos(θ)(cos(θ) + 1)  =  0

Either     cos(θ)  =  0     or     cos(θ) + 1  =  0   --->   cos(θ)  =  -1

                  θ  =  cos^-1(0)     or     θ  =  cos^-1(-1)

So:  θ  =  90°  or  270°  or 180°    (and all 360° increments of these values)

If, by log x², you mean log₁₀x², then:  log₁₀x²  =  4   --->   x²  =  10⁴   --->   x²  =  10,000

--->   x  =  √(10,000)  =  100 

You can use the ( ^ ) key.

Set the expression to y:  y  =  (-3x + 4) / (x - 1)

There are two steps:

Step #1:  interchange the x and y terms   --->   x  =  (-3y + 4) / (y -1)

Step #2:  solve for y:     x(y - 1)  =  -3y + 4

                                    xy - x    =  -3y + 4

                                    xy + 3y  =  x + 4

                                   y(x + 3)  =  x + 4

                                            y  =  (x + 4) / (x + 3)               <---  This is the inverse ...

27^2/3

The numerator of 2 means to square the number. The denominator of 3 means to find the cube root of the number. These can be done in either order.

27^2/3 =  (27²)^1/3  =  729^1/3  =  9

27^2/3 =  (27^1/3)²  =  3² =  9

Multiply any three prime numbers together to get a composite number:

For instance:  3 x 5 x 7 = 105

Can you find two more numbers?

The first location (on the left) will hold one of the 7 math books.

The last location (on the far right) will hold one of the remaining 6 math books.

The middle location (location 6) will hold one of the four history books.

There are 8 books remaining.

In location #2 (from the left), place one of the 8 remaining books.

In location #3, place one of the 7 remaining books.

In location #4, place one of the 6 remaining books.

In location #5, place one of the 5 remaining books.

Location #6 is filled with the previously chosen history book.

In location #7, place one of the 4 remaining books.

In location #8, place one of the 3 remaining books.

In location #9, place one of the 2 remaining books.

In location #10, place the last book.

To find the total number of possibilities, multiply these numbers together:

      7 x  8 x 7 x 6 x 5 x 4 x 4 x 3 x 2 x 1 x 6 

slope  =  m  =  [ y₂ - y₁ ] / [ x₂ - x₁ ]

(x₁, y₁) = (-4, 7)     and     (x₂, y₂) = (-6, -4)

Substitute:  m  =  [ -4 - 7 ] / [ -6 - -4 ]  =  -11/-2  =  11/2

To find the number of packages, divide 84 by 12 ...