I have a question:

why is the angle measurement of a 360 degrees = to 2 pi radians and not 2 pi r radians? Why is pi radians = 180 degrees ? Is pi in this case still 3.141....?

I am confused why we use pi in radians and some times we do not like in the second example provide here Mathwords: Radian.

So confusing >.< !

Think of yourself as being attached to the end of a rope that is nailed to the ground. Obviously as you move, an angle would be swept out. One way to measure the angle would be to measure the proportion of the entire circle that has been swept out. That is the degree measurement, or the pivotal angle.

But another way is to measure the distance you have walked around the outside of the circle, so the arclength. Obviously for different lengths of rope, you would walk a different distance on the arc even if you do sweep out the same angle. So this arclength measurement depends on the length of the radius. So we use our radius as our unit of measurement, and ask ourselves "how many lengths of the radius have we walked along the circumference?" This is what a radian is, a length of the RADIUS on the circumference.

Since every circle has a circumference of \displaystyle \begin{align*} 2\pi \, r \end{align*}, that means that there are exactly \displaystyle \begin{align*} 2\pi \end{align*} lengths of the radius on the circumference, or if you like, \displaystyle \begin{align*} 2\pi \end{align*} radians in a circle.

So the reason we don't use \displaystyle \begin{align*} 2\pi\,r \end{align*} is because that gives us the length of the circumference. But for an angle, all that we want to know is how many lengths of the radius there ARE on the circumference.

Does that make sense?

Hey sakonpure6.

The reason is that radians are the natural unit for angles.

When we define sin(x) we do so in a way that makes sense mathematically in many ways. When we define in radians we get d/dx sin(x) = cos(x) and d/dx cos(x) = -sin(x).

The above helps us obtain the Taylor series expansion for radians and its formula is the simplest it can be.

If we used degrees, then you would get all kinds of effects in differentiation from the chain rule and it would be very messy.

In terms of why pi is used, well that specific number defines the ratio of a circle to its diameter through this relationship among others we get a lot of trigonometric ratios and results linking the trig functions together.

Note that this ratio denotes a frequency much like the frequencies and wave-lengths you are familiar with when you look at a wave.

Ultimately though, the best way to understand it is in terms of projections. If you have a circle of radius 1, then the projection on the x-axis is cos(theta) and the y-axis is sin(theta).

By using the circle/radius relationship you have a total frequency of 2*pi and with all the trigonometrical relationships, you can gauge the significance of pi with respect to projections, natural harmonics (frequencies and wave-lengths of sin(n*pi*x)) and all the ratios and geometry that involve the various trig functions.

With regards to degrees, you have to go back to Babylonian times and understand there base system which was base 60 instead of base 10.

In their base, 60 (minutes, hours, seconds) was a common number and 360 was used as well. This is where we get the degrees from since the babylonians used this in a lot of applications: especially those involving time, cycles, and celestial mechanics (astrology and astronomy).