Question

$\frac{4}{8} = \frac{1}{2tan^{-1}2}$

A & B ratio.

Answer 1

1:2

Answer 2

The given equation is: $\frac{4}{8} = \frac{1}{2\tan^{-1}2}$

A & B ratio.

First, let's analyze the given equation. The left side (LHS) is $\frac{4}{8}$, which simplifies to $\frac{1}{2}$. The right side (RHS) is $\frac{1}{2\tan^{-1}2}$. To check if the equality holds, we need to evaluate $\tan^{-1}2$. Let $x = \tan^{-1}2$. This means $\tan x = 2$. Using a calculator, $\tan^{-1}2 \approx 1.107$ radians. So, the RHS is $\frac{1}{2 \times 1.107} = \frac{1}{2.214} \approx 0.4516$. Since $0.5 \neq 0.4516$, the given equation $\frac{4}{8} = \frac{1}{2\tan^{-1}2}$ is mathematically false.

The question then asks for "A & B ratio." There are no variables A or B explicitly defined in the equation. However, a common interpretation in such cases, especially when a fraction is presented, is to consider the numerator as 'A' and the denominator as 'B' from the most obvious part of the expression that forms a ratio.

From the fraction $\frac{4}{8}$: Let A = 4 Let B = 8

The ratio of A to B is A:B, which is $4:8$. This ratio can be simplified by dividing both numbers by their greatest common divisor, which is 4. $4 \div 4 : 8 \div 4 = 1:2$.

The presence of the false equality involving $\tan^{-1}2$ seems to be a distractor, or a test of critical thinking to identify irrelevant information, as the "A & B ratio" can be directly derived from the numbers 4 and 8.

The core solution is to identify A and B from the fraction $\frac{4}{8}$ and calculate their ratio.

Explanation of the solution:

1. Identify A and B from the fraction $\frac{4}{8}$: A=4, B=8.
2. Calculate the ratio A:B: $4:8$.
3. Simplify the ratio: $1:2$.
4. The truth value of the given equation is irrelevant to finding the ratio of A and B when they are interpreted as the numerator and denominator of the fraction $\frac{4}{8}$.