Understanding the Sin Function

[ehrenfest]

[SOLVED] sin function

Homework Statement

My complex analysis book says, "from calculus, $\sin \theta \leq \theta$ for $0 \leq \theta \leq \pi/2$." Could someone please give me a better reason why that is true?

Homework Equations

The Attempt at a Solution

 

[sutupidmath]

well, the only way i know how to prove that $$|sin(x)|<|x|$$ is with the aid of geometry. YOu need to construct a circle with radius R=1, and denote wit OAB a triangle, and with AB the arch. Denote also by x the angle that the two sides, the radiuses of the circle enclose. And you will see the validity of this. This is true only when the angle is measured in radians. Because also remember that a portion of the arch of the circle, call it acrAB is actually the measure of the angle in radians, if we draw two lines that pass through the two points in the circle A,B.

 

[morphism]

You could also consider the function f(x)=x-sin(x) and it's derivative.

 

[ehrenfest]

You could also consider the function f(x)=x-sin(x) and it's derivative.

Very nice.

 

[Gib Z]

It should also be clear if you use the series definition for sine.

 

[Sourabh N]

Drawing their graphs will also make it clear.

 

[tiny-tim]

… chord-length < arc-length …

Hi ehrenfest!

The intuitive explanation is: 2sinx = chord-length (a straight line ), which is less than 2x = arc-length (not a straight line ), for 2x < π.

(This is sutupidmath's explanation, of course, with the details left out.)

 

[ehrenfest]

It should also be clear if you use the series definition for sine.

Its not clear. For that to be true, you need to know that the sum of all the terms after the first is negative and has magnitude less than the first term in the series. Why is that true?

 

[Sourabh N]

Its not clear. For that to be true, you need to know that the sum of all the terms after the first is negative and has magnitude less than the first term in the series. Why is that true?

Neither to me...

 

[Gib Z]

Its not clear. For that to be true, you need to know that the sum of all the terms after the first is negative and has magnitude less than the first term in the series. Why is that true?

My bad. The things you stated there are sometimes used when deriving the series in the first place and showing the Taylor Remainder term goes to zero, but not always. I wrongly assumed you learned it the same way I did.