State True or False: The minimum value of $n$ for which $ an^{-1}\frac{n}{\pi}\frac{\pi}{4}$, $n\in N$, is valid is 5.

Question

State True or False: The minimum value of $n$ for which $	an^{-1}\frac{n}{\pi}>\frac{\pi}{4}$, $n\in N$, is valid is 5.

Accepted Answer

> [!TLDR]
> The inequality $	an^{-1}\frac{n}{\pi} > \frac{\pi}{4}$ simplifies to $n > \pi$. Since $n$ is a natural number, the smallest such $n$ is 4, not 5. Hence the statement is **False**.

## Concept and Intuition

The core idea here is simple: the inverse tangent function $	an^{-1}x$ is strictly increasing for all real $x$. That means if you want $	an^{-1}(	ext{something})$ to exceed $\frac{\pi}{4}$, that "something" must be greater than the value whose tangent is exactly $\frac{\pi}{4}$.

What value gives $	an^{-1}x = \frac{\pi}{4}$? It's $x = 1$, because $	an\frac{\pi}{4} = 1$. So the inequality $	an^{-1}\frac{n}{\pi} > \frac{\pi}{4}$ is equivalent to $\frac{n}{\pi} > 1$, provided we are careful about the domain (which is all real numbers here, so no issues).

Once we have $n > \pi$, the smallest natural number $n$ satisfying this is $n = 4$, since $\pi \approx 3.14$. The statement claims it's 5 — that's off by one.

## Step-by-Step Solution

1. **Set up the inequality**  
   We are given:  
   $$	an^{-1}\frac{n}{\pi} > \frac{\pi}{4}, \quad n \in \mathbb{N}$$

2. **Apply the monotonicity of $	an^{-1}$**  
   The function $	an^{-1}x$ is strictly increasing on $\mathbb{R}$. Therefore, for any $a, b$:  
   $$	an^{-1}a > 	an^{-1}b \iff a > b$$  
   Here, take $a = \frac{n}{\pi}$ and $b = 1$ (since $	an^{-1}1 = \frac{\pi}{4}$).  
   So the inequality becomes:  
   $$\frac{n}{\pi} > 1$$

3. **Solve for $n$**  
   Multiply both sides by $\pi$ (positive, so inequality direction stays):  
   $$n > \pi$$

4. **Find the smallest natural number**  
   $\pi \approx 3.14159\ldots$ …

Q.State True or False: The minimum value of $n$ for which $\tan^{-1}\frac{n}{\pi}>\frac{\pi}{4}$ , $n\in N$ , is valid is 5.

Concept understanding — Inverse Tangent Identity

Inverse Tangent Identities

The core addition identity

Subtraction

The doubling identity

The complementary identity

Concept and Intuition

Step-by-Step Solution

Method: Solve an inverse-trig inequality via monotonicity

Steps

Common Mistakes

The core idea: the range of $\tan^{-1}$ and the product condition

Applying it to this problem

Q.State True or False: The minimum value of nnn for which tan⁡−1nπ>π4\tan^{-1}\frac{n}{\pi}>\frac{\pi}{4}tan−1πn​>4π​, n∈Nn\in Nn∈N, is valid is 5.

Concept and Intuition

Step-by-Step Solution

🎓Unlock everything free for 14 days

Q.State True or False: The minimum value of $n$ for which $\tan^{-1}\frac{n}{\pi}>\frac{\pi}{4}$ , $n\in N$ , is valid is 5.

Unlock everything free for 14 days