Is it Possible to Prove this Trigonometric Inequality?

In summary, the "Trig Inequality Challenge" is a mathematical exercise that involves solving inequalities with trigonometric functions. It is important for developing understanding and skills in trigonometry and has real-world applications. Common strategies for solving these problems include using identities and graphing, while tips for efficiency include understanding the properties of trigonometric functions and breaking down the problem. Some common mistakes to avoid include not considering domain and range, not simplifying fully, and making sign errors. It is also important to pay attention to any restrictions on variable values.
  • #1
lfdahl
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Prove the inequality:

\[\left | \cos x \right |+ \left | \cos 2x \right |+\left | \cos 2^2x \right |+...+ \left | \cos 2^nx \right |\geq \frac{n}{2\sqrt{2}}\]

- for any real x and any natural number, n.
 
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  • #2
Hint:

What is the range of the function:$f(x) = |\cos x| + |\cos 2x|$?
 
  • #3
Suggested solution:

Note, that:

\[\left | \cos x \right |+\left | \cos 2x \right |=\left | \cos x \right |+\left | 2\cos ^2x-1 \right | \equiv t + \left | 2t^2-1 \right |\geq \frac{1}{\sqrt{2}}, \: \: \: \: 0\leq t\leq 1.\]

For odd values of $n$, we get $k+1$ pairs ($n = 2k+1$):
\[\left ( \left | \cos x \right |+\left | \cos 2x \right | \right )+\left ( \left | \cos 4x \right |+\left | \cos 8x \right | \right )+ ...+\left ( \left | \cos 2^{n-1}x \right |+\left | \cos 2^{n}x \right | \right )\geq \frac{k+1}{\sqrt{2}}=\frac{n+1}{2\sqrt{2}} > \frac{n}{2\sqrt{2}}\]

For even values of $n$ ($n = 2k+2$):

\[\left | \cos x \right |+\left ( \left | \cos 2x \right | + \left | \cos 4x \right |\right )+ \left ( \left | \cos 8x \right | +\left | \cos 16x \right | \right ) ...+\left ( \left | \cos 2^{n-1}x \right |+\left | \cos 2^{n}x \right | \right ) \geq =\frac{n}{2\sqrt{2}}\]
 

FAQ: Is it Possible to Prove this Trigonometric Inequality?

What is the "Trig Inequality Challenge"?

The "Trig Inequality Challenge" is a mathematical exercise that involves solving inequalities with trigonometric functions, such as sine, cosine, and tangent.

Why is the "Trig Inequality Challenge" important?

The "Trig Inequality Challenge" is important because it helps students develop their understanding of trigonometric functions and their ability to solve complex mathematical problems. It also has real-world applications in fields such as engineering, physics, and astronomy.

What are some common strategies for solving "Trig Inequality Challenge" problems?

Some common strategies for solving "Trig Inequality Challenge" problems include using trigonometric identities, graphing the functions, and using algebraic manipulation techniques. It is also important to carefully consider the domain and range of the trigonometric functions involved.

What are some tips for solving "Trig Inequality Challenge" problems more efficiently?

Some tips for solving "Trig Inequality Challenge" problems more efficiently include understanding the properties of trigonometric functions, practicing regularly, and breaking down the problem into smaller, more manageable parts. It is also helpful to check your solutions by plugging them back into the original inequality.

Are there any common mistakes to avoid when solving "Trig Inequality Challenge" problems?

Some common mistakes to avoid when solving "Trig Inequality Challenge" problems include forgetting to consider the domain and range, not simplifying expressions fully, and making sign errors when manipulating inequalities. It is also important to pay attention to any restrictions on the variable values.

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