How to Establish the Inequality xcosx < sinx < x Using the Unit Circle?

Click For Summary
SUMMARY

The inequality xcosx < sinx < x for 0 < x < π/2 can be established using the unit circle. This proof is a well-known mathematical challenge often encountered in calculus and trigonometry courses. The provided resource, a PDF from California State University Northridge, outlines the first part of the proof, which is essential for understanding the relationships between the sine and cosine functions in this context. The discussion confirms that this inequality is a classic problem that many can solve.

PREREQUISITES
  • Understanding of trigonometric functions, specifically sine and cosine.
  • Familiarity with the unit circle and its properties.
  • Basic knowledge of inequalities in calculus.
  • Experience with mathematical proofs and problem-solving techniques.
NEXT STEPS
  • Review the PDF resource on sine inequalities from California State University Northridge.
  • Study the properties of the unit circle in relation to trigonometric functions.
  • Practice establishing other trigonometric inequalities using graphical methods.
  • Explore advanced calculus concepts related to limits and continuity in trigonometric functions.
USEFUL FOR

Mathematics students, educators, and anyone interested in deepening their understanding of trigonometric inequalities and proofs.

theperthvan
Messages
182
Reaction score
0
This was an exam question I had at the end of 2005 in my uni entrance exams.
Can you do it?

Establish the inequality

[tex]xcosx < sinx < x[/tex] where [tex]0<x<pi/2[/tex] using the unit circle.

is the tex working?
 
Physics news on Phys.org
This is a classic. Here:

http://www.csun.edu/~ac53971/courses/math350/xtra_sine.pdf"

The first part of the proof is what you are looking for.

Edit: Is it a challenge? If so then most people here will be able to do it, this is a very known proof.
 
Last edited by a moderator:
That was quick!

Yes, that's it. Cheers,
 

Similar threads

Undergrad How to know if a complex root is inside the unit circle
  • · Replies 5 ·
Replies
5
Views
3K
High School Question about change of variables
  • · Replies 29 ·
Replies
29
Views
5K
MHB How Much Should the Radius Increase to Enlarge the Circle's Area by b Units?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Inequality with integral and max of derivative
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Integrating Sin(x): Solve the Area of a Half Circle
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Inequality - how to prove that xcosx < sinx for all x
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Solving Trig Integrals with Residue Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Finding a unit normal to a surface
  • · Replies 6 ·
Replies
6
Views
2K
Circle approximation of a wave pulse on a string or spring
  • · Replies 29 ·
Replies
29
Views
2K