Understanding Trigonometry (WIP)

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Pi

When we roll out the circumference (perimeter) of a circle into a straight line and then measure how many times we can fit the diameter in that line we will always get 3.14…So 3 diameters and a bit, .14..., make up the circumference. This is the same as dividing the circumference by the diameter.

> [ What is Pi and Where Does it Actually Come From? | Part 1 ](https://www.youtube.com/watch?v=XKkBDWP3IWA)

Using algebra we can find out the circumference by taking 3.14 or pi and multiplying it bythe diameter to cancel out the division, e.g.

<LatexSplit c e=” \dfrac{\text{Circumference}}{\text{Diameter}} &= \pi \[1em]

\dfrac{\text{Circumference}}{\cancel{\text{Diameter}}} \cdot \cancel{{\color{red}{\text{Diameter}}}} &= \pi \cdot {\color{red}{\text{Diameter}}} \[1em]

\text{Circumference} &= \pi \cdot \text{Diameter} \[1em] ”/>

Pi is an irrational number because there is no repeating pattern in the decimals (e.g., 0.333333... = 1\3) that enable it to become a fraction and it’s irregularity continues forever. That’s why the greek symbol is used as a placeholder!

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Trig

Radians

Radians are a measure of angles that are defined by circles.

When we think of a full 360 degree rotation we can describle that as

Let say the circumference is 2 and the radius is 1

We will say

Therefore, and

And so radians describe the relationship of fractional pi to angles!

E.g,

You may notice that there is no symbol like w/ and that’s because the symbol.

Radians are a dimensionless quantity. They are the ratio between the arc length and the radius of the circle (hence they’re not actually units). Ratios are just a number with no unit of measurement attached. To go 1 radian in rotation means to go 1 radius around the circumference, as an arc length. 1 radian is the angle that cuts out an arc equal to the radius.

Trigometric Functions

The trig functions sine, cosine and tangent are functions meaning it’s a mapping from input to output and not something that holds the distrubitive property, e.g., sin(x+y) != sin(x) + sin(y). The same goes with logarithms.

When thinking about 360 degrees for a full circle that number comes from the 360 days it takes for the Earth to do a full revolution around the sun.

We can think of the trigometric functions of sin, cos and tan as co-ordinates on a circle. Therefore, if we were to do a full rotation of 360 degrees on top of our current angle of 45 degrees we would get 405 degrees. BUT the result of any of the trigometric functions w/ 45 or 405 degrees will be the exact same!

If we create a 2d graph of each co-ordinate on a circle using the values provided from sin, cos, and tan we’ll notice that sin and cos are the exact same with the caveat that sin’s x starting position is translated to the right.

Since sin is graphing the y axis when we go counter clockwise from x = 1, y = 0 then our . When as we increase by 45 degrees then we’ll see a pattern

This is interesting, so it’s saying the y axis is 0.5 tall.

When we move to the peak, 90 degrees, We reach the absolute peak of the circle and as we continue to rotate,

The y axis is still positive but is the exact same as 45 degrees. Interesting…

Then what would happen for

And then following this pattern the bottom point of the circle

So we can draw this graph out from 0 degrees to 360 degrees and then it continuously copies the same pattern from 0-360 starting at 360 as the new 0, i.e., it oscilates. We actually call it a sinusodial wave.

We see cosine start at 1 because the x is already at the peak at . It’s the complement of the sine graph, its just off by 90 degrees.

Tangent on the other hand is quite interesting. This is when we get intoduced to limits,