Write 'True' or 'False' and justify your answer in each of the following:
The value of $ 2 \sin \theta $ can be $ a+\frac{1}{a} $, where $ a $ is a positive number, and $ a
eq 1 $

Given:

The value of \( 2 \sin \theta \) can be \( a+\frac{1}{a} \), where \( a \) is a positive number, and \( a ≠ 1 \).

To do:

We have to find whether the given statement is true or false.

Solution:

$a$ is a positive number and $a ≠1$

This implies,

$AM>GM$

AM and GM of two numbers $a$ and $b$ are $\frac{(a+b)}{2}$ and $\sqrt{a b}$.

Therefore,

$\frac{a+\frac{1}{a}}{2}>\sqrt{a \times \frac{1}{a}}$

$(a+\frac{1}{a})>2$

$2 \sin \theta>2$      ($2 \sin \theta=a+\frac{1}{a}$)

$\sin \theta>1$ which is not possible.        [Since $-1 \leq \sin \theta \leq 1$]

Hence, the value of $2 \sin \theta$ cannot be $a+\frac{1}{a}$.

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Updated on: 10-Oct-2022

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