In high school, you learned about the square root and the **sine wave**s. Both of these functions are simple, but the square root is simple. The sine wave is more complicated, but not impossible!

Both the square root and the sine wave have a place in mathematics and science, so it does not make much of a difference which function you choose to represent. Both are valid ways to measure radiance.

The only difference between the two measures is how they convert from a number to a name – the square root is r = sqrt(1+i), and the sine wave is w = cos(1+i) – cose(1+i).

When talking about Radiance Measurements, it is important to use either the Square Root or Sine Wave Method.

## i6 – 4i4

In trigonometry, the angle i between a line and its adjacent sides is called the angle i. The angle i represents a 90-degree angle, so it is a good place to start.

The expression 4i4 represents a quantity of 4 units, or ⅛ inch. This expression represents the size of an inch of something.

In this case, it represents the size of an inch of space. If you measure an inch in this space, you *get ¼ inch* of space. If you measure an inch inside that space, then you have gone outside that space!

This is why we say that there are four inches in a foot: There are *four steps inside* each foot (or take off your shoes and look at what you are measuring).

To find out if **i6 – 4i4** is equivalent to F(i**), try dividing** both sides by 2 and see if your answer is equivalent to F(i).

## i9 – 6i5

When we look at the graph of a function, we can find the equivalent equation to determine when x = 0. In order for us to do this, we must know the value of x at any point in time.

The value of *x changes depending* on when the function is applied. For example, when putting a mattress pad between your feet and sleeping on a soft surface, the value of x changes as you move from one side to the other due to different surfaces.

When applying a f(x) = 5 expression to an i(x) = 6 expression, we get an *i9 – 6i5 equation*. This means that **5 – 6 yields 9** and *6 yields 5*. When solving these equations, we get values of 5 and 6 that are closest to our original input so that it matches what we are looking for.

## i12 – 8i6

When i is an integer, the *following two expressions* are equivalent:

i12 – 8i6

i12 + 8i6

The only difference between these two expressions is the number i. If i is an integer, then the second expression is equivalent to the first! For example, 4 corresponds to i4, not 4 + 8.

If you look at the second expression, it doesn’t seem too difficult until you realize that it takes a long time to solve for x. By using a shorter expression that *takes less time* to solve for x, F(x) can be calculated in less time!

This is why *many computer algorithms* will combine the two expressions into one when i is an integer.

## i15 – 10i7

In this case, i is an integer, and x is a number. The number x is the value of i times the variable x.

The variable i is the index of the row in the matrix that holds the variable x.

So, if v = (x1 + x2 + … + X) ** – 1**, then v(i) = (x1 + x2 + … + X) – 1, which is a number.

By definition, a vector can have only **one value** in its component values. That being said, we can ask whether or not v has **another value equivalent** to 0 within our equation for v.

## i18 – 12i8

If F(i) = *x3 – 2×2*, Which Expression Is Equivalent to F(i+1)?

We can use this to our advantage. If we set the variable name to Äì and compare the two equations: Äì == 3×2 + 4, Then we know that the first equation does not equal 3×2 + 4, which tells us that *i18 – 12i8 == 0* is true. We can then substitute in our value for x and see if it works. It does!

If we substituted in 10 for x and *wrote 3×2* + 10 = 18 + 12 20 = 18 + 12 which agrees with our original equation of 3×2 + 10 = 18 + 12 19. This proves our hypothesis that 9

## i21 – 14i9

If F(i) = x3 – 2×2, Then the equivalent expression for i is ** i21 – 14i9**. This is equivalent to F(i) = x3 +

*2×2 – 14×9*.

This can be written as a quadratic equation: i21 – 14i9 = x3 + 2×2 – 14×9, which can be rearranged to get x3 + 2×2 – 14×9.

In fact, this equation has two solutions: 1) i = 0 and 2) i = 19. When i = 0, then F(0) = 1 and F(19) = 16 because they are both positive. When i 1 because they are both negative.

When **solving quadratic equations like** this one, it is important to pay attention to whether or not the inequality is on either side of the equation.

## i24 – 16i10

The i and the 24 represent an index into the matrix. The 3 and the 10 represent a matrix row and column. We are looking for a row or column value that is equal to 16.

This can be tricky at first, but don’t worry! You will get it in a minute.

First, notice that when we *multiply one matrix* by another, the **new matrix gets** its own value for one of its dimensions. This dimension is called the vectorial basis of the new matrix.

The vectorial basis of a mathematically defined X3 – Y3 – Z3 – Matrix is:

We can use this to find our matching i24 – 16i10 X3 – 2×2 Y3 – 2×2 Z3 – Matrix.

## i27 – 18i11

When i is equal to x, the equation F(i) = F(x) **– 2×2** is equivalent to F(i) – 18x + *18 – 2×2*, which is **i27 – 18i11**.

This happens when a **small number appears** in both sides of the equation. For example, 18 appears once on the left side and once on the right side of the equation.

Similarly, 21 appears on the left side and 9 appears on the right side of the equation.