Trigonometric inequality problem.

AI Thread Summary
The discussion revolves around proving the trigonometric inequality 0 ≤ (1 + sin(x)) / (5 + 4cos(x)) ≤ 10/9 for all x. Participants suggest sketching the graph of the function to visualize its behavior and inquire about methods to determine its maximum and minimum values. There is a focus on ensuring the denominator is never zero or negative, which is crucial for the inequality's validity. One proposed approach is to analyze the critical points and prove that the expression 1 + sin(x) - (10/9)(5 + 4cos(x)) remains negative. The conversation emphasizes the importance of both the numerator and denominator in establishing the bounds of the function.
Michael_Light
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Homework Statement



Deduce that 0 ≤
MSP154219h2f53c89aa97ch00001f2200631f7c6f1g.gif
≤ 10/9 for all values of x.

Homework Equations


The Attempt at a Solution



Is it possible to sketch a graph for
MSP154219h2f53c89aa97ch00001f2200631f7c6f1g.gif
? How?

Or is there any methods to find the max./min. value of
MSP154219h2f53c89aa97ch00001f2200631f7c6f1g.gif
?

Please enlighten me...
 
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is the denominator ever zero or negative? if not try mutiplying through by it

or find the critical points to fiind max and min
 
If you can prove 1+sin(x)-\frac{10}{9}(5+4cos(x)) is always less than 0, you can get the less than part. If you think about the numerator, the greater than part also follows quickly.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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