What does my calculator do to get -0.24192189 when I enter in Cos(256)?

Question

In my quest to better understand trigonometry, It would be cool for someone to manually (dont use a calculator) calculate cos(256) and show me how they did it, include every step :)

Don't explain to me what cos does, just work it out and show me every step, I'll try to figure it out on my own.

Thanks

Answer 1

Well, it's nearly impossible to work this out manually. I can explain why the answer is -o.24.....

Image a graph with the x- axis and y-axis. There is a circle this a radius of 1 and is positioned at the center of the circle. Now, cos
is the x axis and sin is the y axis( but u don't have to worry about the y). When the circle is 256 degrees going anti clock wise from positive 1on the x axis, the x axis will equal -0.24...

That's the best I can do ...good luck!

Answer 2

I don't think it will be very helpful in your trig study because the method that computers use to give values to functions is much different than the way we might go about it, but here goes:

cos(x)

Examine x.... is it larger than 2pi or less than zero? Take the number divided by 2pi and take the remainder, this is the angle it uses

Look here for an explanation of the some the finer points of the calculation here: http://www.math.niu.edu/~rusin/known-math/94/cordic

It probably has what is called a "trig look up table" Essentially, someone calclulated (by hand) certain special angles for the cos, sin, exp, etc, a loooong time ago and they're programmed (to a high accuracy) straight into the memory of the calculator. Using trig addition identities, it can add up to the desired angle quite quickly.

If it were the case that you could not use the method above, there is another method by approximation of Taylor polynomial series. You can expand the cos like so:

cos(x)=sum_(i=0...infty) (-1)**2x**2/(2*i)!!

Choosing a convenient scaling of x (ie, less than one) assures this will converge to a value to a desired approximation after a certain number of iterations, then you scale the answer back after your done. This is less efficient.

If you didn't have a calculator (ie, about thirty years ago) you used the CRC math tables which have most values of the elementary functions up to about two decimal places.

Answer 3

The cos, sin and tan ratios can be calculated using the unit circle. This is a circle drawn on the x/y plane around the origin (0,0) with a radius of 1 unit. The cos ratio refers to the x value of the point on the circle which corresponds to angle made from the origin (in your case 256). So when you find the angle of 256 degrees (from the origin and positive half of x plane) the x value on the circle is -0.24192189. This can be so for example if you take cos(90)=0 cos(180)=-1. The same is for the Sin ratios, except it is the y value not the x value. And the Tan ratio is the corresponding y value divided by the corresponding x value (y/x). Try googling Unit Circle trigonometry

Answer 4

Calculators use Taylor series to approximate trig functions.
It pretty much requires calculus to understand how to get these series. But if you check about the middle of the page of the source I cited, they give the series for the trig functions.

Answer 5

This does not answer youre question, but it might be useful. Calculators, computating machins use trig. look-up tables which are not quite accurate. e.g. cos, sin x deg. does not produce 0.e.g. 90.