Unit Circle & Trig: Explaining Why sin(t+2pi)=sin t

[haribol]

In a unit circle, where t is a real number, why is that the following is true:

sin (t+2pi)=sin t

I really don't understand this, if I put any value into t and check on my calculator, sint and sin(t+2pi) give different answer, why is this?

Thanks for any help

 

[cdhotfire]

think of what 2pi represents in a unit circle.

 

[Pyrrhus]

sin (t+2pi), well that is equal to sint*cos2pi + cost*sin2pi, cos2pi=1 and sin2pi=0 so that's equal sint

 

[cdhotfire]

i think that explanition will confuse him, all he has to no is what 2pi represents if we he is talking about radians and he will understand.

 

[Pyrrhus]

i think that explanition will confuse him, all he has to no is what 2pi represents if we he is talking about radians and he will understand.

You're right..

 

[cdhotfire]

thank you

 

[Integral]

Perhaps your calculator is set for degrees rather then radians. If you cannot figure out how to change it to radians you may try sin(t) and sin(t+360).

 

[cdhotfire]

That would work also.

 

[HallsofIvy]

First, forget about right triangles. That definition of "sine" and "cosine" is only true for angles between 0 and 90 degrees and doesn't work here.

A more general definition of sine and cosine is this: Start at the point with coordinates (1,0). Now, for t>= 0, measure, counterclockwise, around the unit circle a distance t: the coordinates of the point you end at are, by definition, (cos(t), sin(t)) (If t is negative, then measure clockwise).

Now, that's the UNIT circle: it has radius 1 and diameter 2- therefore circumference 2π. To find sin(t+ 2π), you would measure around the circle a distance t, then an additional 2π. Since that additional distance is exactly the circumference of the circle, it takes you right back to the original point so that sin(t+ 2π) and sin(t) give exactly the same thing: the y coordinate of the point (and cos(t+ 2π) and sin(t) are both equal to the x coordinate).

Notice that t, in that definition, isn't an angle at all! It is a measure along the circumference of the circle. However, calculators are not designed by mathematicians, they are designed by engineers and engineers think of sine and cosine in terms of angles! The "radian" is defined so that the angle, in radians, is exactly the same as the distance around the unit circle.

That's the reason you calculator is not giving the "correct" answer is that your calculator is in "degree" mode rather than "radian" mode. Some calculators have a "degree, radian, grad" or "d,r,g" key for changing from one to the other. Some have a "mode" key that allows you to change a variety of things including "radian" or "degree" mode.

You should understand that, basically, the only time you use "degrees" in when you are working with actual angles that are measured in degrees. When you are working with the trig functions as actual "functions", you always use "radians".

 

[haribol]

I understand now, thanks guys

 

[cdhotfire]

np, take it ez.