# Trigonometric inequality: sin (1/(n+1934))<1/1994

• MHB
• anemone
In summary, a trigonometric inequality involves trigonometric functions and can be solved using algebraic techniques and the properties of trigonometric functions. It is true for all values of n if there are no values of n that make the inequality false. To check the solution, you can plug in different values for n or graph the inequality.
anemone
Gold Member
MHB
POTW Director
Find the smallest natural number $n$ for which $\sin \left(\dfrac{1}{n+1934}\right)<\dfrac{1}{1994}$.

As $\sin x<x$ for all $x>0$ – in particular $\sin\left(\dfrac1{1994}\right)<\dfrac1{1994}$ – it suffices to show that $\sin\left(\dfrac1{1993}\right)>\dfrac1{1994}$.

By Taylor’s theorem, $\sin x=x-\dfrac{x^2}2\sin\xi$ for some $0<\xi<x$ (using the Lagrange form of the remainder). Thus:

$\begin{array}{rcl}\sin x &=& x-\dfrac{x^2}2\sin{\xi} \\\\ {} &>&x-\dfrac{x^2}2\xi \\\\ {} &>& x-\dfrac{x^3}2.\end{array}$

Hence $\sin\left(\dfrac1{1993}\right)>\dfrac1{1993}-\dfrac{\left(\frac1{1993}\right)^3}2 = \dfrac{7944097}{15832587314}>\dfrac1{1994}$, as required. So the smallest natural number is $\boxed{n=60}$.

Code:
    for (unsigned int n = 0; n <= 60; n++)
if ((sin(1/(n + 1934.0))) < 1/1994.0)
std::cout << n;

(Bigsmile)

anemone said:
Find the smallest natural number $n$ for which $\sin \left(\dfrac{1}{n+1934}\right)<\dfrac{1}{1994}$.
[sp]
For $0<x<1$, we have $\sin(x)<x$. This shows that the inequality is satisfied for $n=60$. Taking $n=59$, we find:
$$\sin\left(\frac{1}{1993}\right)\approx 0.00050176> \frac{1}{1994}\approx 0.00050150$$
As $\sin\left(\dfrac{1}{n+1934}\right)$ is a decreasing function of $n$ for $n>0$, this shows that the smallest $n$ that satifies the inequality is $60$.
[/sp]

## 1. What is a trigonometric inequality?

A trigonometric inequality is an inequality that involves trigonometric functions, such as sine, cosine, and tangent.

## 2. What does the inequality sin(1/(n+1934))<1/1994 mean?

This inequality means that the value of sine of 1 divided by n plus 1934 is less than 1 divided by 1994.

## 3. How do I solve a trigonometric inequality?

To solve a trigonometric inequality, you need to use algebraic techniques and the properties of trigonometric functions to isolate the variable and determine its possible values.

## 4. What does it mean if the inequality is true for all values of n?

If the inequality is true for all values of n, then it means that the statement is always true, regardless of the value of n. In other words, there are no values of n that make the inequality false.

## 5. How can I check if my solution to a trigonometric inequality is correct?

You can check your solution by plugging in different values for n into the original inequality and seeing if the inequality holds true for each value. You can also graph the inequality and see if your solution falls within the shaded region.

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