Inequality involving theta and sin(theta)

In summary, the conversation discusses a homework problem involving proving the inequality \frac{2}{\pi} \theta \leq\sin\theta for 0\leq \theta\leq\frac{\pi}{2}. The person has attempted to use a limit and compare the growth rates of \sin\theta and \theta, but has not been successful. They suggest using calculus to show that the graph of \sin\theta is concave down on the given interval, which would prove the inequality.
  • #1
Sistine
21
0

Homework Statement


I'm trying to prove the following inequality for [tex]0\leq \theta\leq\frac{\pi}{2}[/tex]

[tex] \frac{2}{\pi} \theta \leq\sin\theta[/tex]


Homework Equations



[tex] 0 \leq \theta\leq\frac{\pi}{2} [/tex]

[tex] 0\leq \frac{2}{\pi}\theta\leq 1 [/tex]

The Attempt at a Solution


I've looked at the limit

[tex] \lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1 [/tex]

I've also looked at other inequalities involving [tex]\sin\theta[/tex]

[tex]\sin\theta<\theta[/tex]

I've also tried to take derivatives to see how fast [tex]\sin\theta[/tex] grows in comparison to [tex]\theta[/tex], but I have not managed to prove the inequality.
 
Physics news on Phys.org
  • #2
That limit won't do you much good, since it is for x near zero.

The graphs of y = 2x/[itex]\pi[/itex] and y = sin(x), you'll see that the sine curve is above the line, and that the two intersect at the origin and at ([itex]\pi[/itex], 1), and at no other points on the interval you're interested in. This isn't a proof, though, but you should be able to convey what the graph shows through calculus.

You should be able to establish your inequality by showing that the graph of y = sin(x) is concave down on the interval (0, [itex]\pi[/itex]), meaning that the graph of y = sin(x) will be above the graph of y = 2x/[itex]\pi[/itex] except at the endpoints of your interval.
 

1. What is the definition of theta?

Theta is a Greek letter commonly used in mathematics to represent an angle.

2. How is sin(theta) related to theta?

Sin(theta) is a trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle with an angle of theta.

3. Can theta be negative?

Yes, theta can be negative as it represents an angle and angles can be measured in both clockwise and counterclockwise directions.

4. What is the range of values for theta?

Theta can take on any real value between 0 and 360 degrees or 0 and 2π radians, depending on the unit of measurement.

5. How does inequality involving theta and sin(theta) arise in scientific studies?

Inequality involving theta and sin(theta) can arise in scientific studies when analyzing data or models that involve angles and trigonometric functions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
920
  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus and Beyond Homework Help
Replies
6
Views
274
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
837
  • Calculus and Beyond Homework Help
Replies
28
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
742
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
253
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Back
Top