Trigonometric inequality bounded by lines

In summary, the conversation discusses a problem involving an inequality with trigonometric expressions in the context of damped motion in spring-mass systems. After some algebraic manipulation, the inequality can be simplified using a linear-combination identity and a vector dot product. The validity of the last inequality is shown to hold on a specific range of x values.
  • #1
kalish1
99
0
How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$

This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here.

Here is the illustrative graph provided by Wolfram Alpha:

[1]: http://i.stack.imgur.com/oWf9E.png

Thanks!
 
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  • #2
I would use a linear-combination identity to obtain the amplitude $A$ of the trigonometric expression:

\(\displaystyle A=\sqrt{(16x)^2+(4x-2)^2}<\sqrt{(17x)^2}\)

What do you find?
 
  • #3
The LHS can be expressed as a vector dot product:

\[16xcos(8x)+4xsin(8x)-2sin(8x)=16xcos(8x)+(4x-2)sin(8x)=\\\\ \binom{16x}{4x-2}\cdot \binom{cos(8x)}{sin(8x)}=\vec{a}\cdot \vec{e}\\\\ Applying\; the \; Cauchy-Schwarz \; inequality:\\\\ \left |\binom{16x}{4x-2}\cdot \binom{cos(8x)}{sin(8x)} \right |\leq \left \| \binom{16x}{4x-2} \right \|=\sqrt{(16x)^{2}+(4x-2)^2}\leq \sqrt{(17x)^{2}}\]

My question: How do you show the validity of the last inequality, if we don´t know the range of x:

\[\sqrt{(16x)^{2}+(4x-2)^2}\leq \sqrt{(17x)^{2}}\;?\]
 
  • #4
We can show that this inequality is true on:

\(\displaystyle \left(-\infty,\frac{2\left(-4-\sqrt{33} \right)}{17} \right)\,\cup\,\left(\frac{2\left(-4+\sqrt{33} \right)}{17},\infty \right)\)
 
  • #5


I would approach this problem by first understanding the context in which it arises - damped motion in spring-mass systems. This helps us to visualize the physical meaning behind the inequality and gives us a starting point for solving it.

Next, I would use my knowledge of trigonometric functions and their properties to simplify the equation. For example, we can use the identity $cos^2(x)+sin^2(x)=1$ to rewrite $16x\cos(8x)+4x\sin(8x)-2\sin(8x)$ as $16x\cos(8x)+2x\sin(8x)$. This simplification may help us to better understand the behavior of the inequality.

We can also use the fact that $|17x|=17x$ if $x\geq 0$ and $|17x|=-17x$ if $x<0$. This gives us two different cases to examine: when $x\geq 0$ and when $x<0$.

For the case of $x\geq 0$, we can use the properties of cosine and sine functions to show that $16x\cos(8x)+2x\sin(8x)$ is always less than or equal to $17x$. This can be seen by noting that the maximum value of $\cos(8x)$ is $1$ and the maximum value of $\sin(8x)$ is $1$ when $x=0$. Therefore, the left side of the inequality is always less than or equal to $17x$, which satisfies the inequality.

For the case of $x<0$, we can use similar reasoning to show that $16x\cos(8x)+2x\sin(8x)$ is always greater than or equal to $-17x$. This can be seen by noting that the minimum value of $\cos(8x)$ is $-1$ and the minimum value of $\sin(8x)$ is $-1$ when $x=0$. Therefore, the left side of the inequality is always greater than or equal to $-17x$, which also satisfies the inequality.

In conclusion, we have shown that for all values of $x$, the left side of the inequality is either less than or equal to $17x$ or greater than or equal to $-17x$, which satisfies the inequality. Therefore
 

FAQ: Trigonometric inequality bounded by lines

What is a trigonometric inequality bounded by lines?

A trigonometric inequality bounded by lines is an inequality that involves trigonometric functions (such as sine, cosine, and tangent) and is bounded by linear equations or inequalities. This means that the variables in the inequality are constrained by a set of lines or linear equations.

How do I solve a trigonometric inequality bounded by lines?

To solve a trigonometric inequality bounded by lines, you need to first isolate the trigonometric function on one side of the inequality. Then, you can use algebraic techniques to solve for the variable. Finally, you can use the properties of trigonometric functions to find the solutions that satisfy the inequality within the given bounds.

What is the importance of trigonometric inequalities bounded by lines?

Trigonometric inequalities bounded by lines are important in mathematics because they allow us to analyze and understand the behavior of trigonometric functions within a specific interval. This can be useful in various fields such as physics, engineering, and geometry.

What are some common techniques used to solve trigonometric inequalities bounded by lines?

Some common techniques used to solve trigonometric inequalities bounded by lines include using trigonometric identities, graphing the functions and their bounds, and applying algebraic manipulations such as factoring and completing the square.

Can trigonometric inequalities bounded by lines have multiple solutions?

Yes, trigonometric inequalities bounded by lines can have multiple solutions. This is because trigonometric functions are periodic, meaning they repeat their values over certain intervals. Therefore, there can be multiple points within the given bounds that satisfy the inequality.

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