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Question:
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Let $\theta$ be a positive angle, If the number of degrees in $\theta$ is divided by the number of radians in $\theta$, then an irrational
number $\frac{180}{\pi}$ results. If the number of degrees in $\theta$ is
multiplied by the number of radians in $\theta$, then an irrational
number $\frac{125 \pi}{9}$ results. The angle $\theta$ must be equal to
Options:
number $\frac{180}{\pi}$ results. If the number of degrees in $\theta$ is
multiplied by the number of radians in $\theta$, then an irrational
number $\frac{125 \pi}{9}$ results. The angle $\theta$ must be equal to
Solution:
1587 Upvotes
Verified Answer
The correct answer is:
$50^{\circ}$
From going by the options, option (a), $\theta=30^{\circ}$, as we know that $180^{\circ}=\pi \mathrm{radian}$
$\therefore \quad 30^{\circ}=\frac{30 \pi}{180}$ radian
Now according to question,
$\frac{30^{\circ} \times 180^{\circ}}{30^{\circ} \pi}=\frac{180}{\pi}$
Now number of degree in $\theta$ is multiplied by number of radians in $\theta$.
$\therefore \quad 30^{\circ} \times \frac{30 \pi}{180}=\frac{900 \pi}{180}=\frac{10 \pi}{2}=5 \pi \neq \frac{125 \pi}{9}$
From option (b), $\theta=45^{\circ}$
$\therefore \quad 45^{\circ}=\frac{45^{\circ} \pi}{180}$ radian
Now according to question,
$\frac{45^{\circ} \times 180}{45^{\circ} \pi}=\frac{180}{\pi}$
Now number of degree in $\theta$ is multiplied by number of $\operatorname{radian}$ in $\theta$.
$\therefore \quad 45^{\circ} \times \frac{45^{\circ} \pi}{180^{\circ}}=\frac{45 \pi}{4} \neq \frac{125 \pi}{9}$
From option (c), $\theta=50^{\circ}$
As we know that $180^{\circ}=\pi$ radian
$\therefore \quad 50^{\circ}=\frac{50 \pi}{180}$ radian
Now according to question
$\frac{50^{\circ} \times 180^{\circ}}{50^{\circ} \pi}=\frac{180}{\pi}$
Now number of degree in ' $\theta$ ' is multiplied bynumber of $\operatorname{radian} \operatorname{in} \theta .$
$\therefore \quad 50^{\circ} \times \frac{50 \pi}{180}=\frac{2500 \pi}{180}=\frac{125 \pi}{9}$
$\therefore \quad$ Option (c) is correct.
$\therefore \quad 30^{\circ}=\frac{30 \pi}{180}$ radian
Now according to question,
$\frac{30^{\circ} \times 180^{\circ}}{30^{\circ} \pi}=\frac{180}{\pi}$
Now number of degree in $\theta$ is multiplied by number of radians in $\theta$.
$\therefore \quad 30^{\circ} \times \frac{30 \pi}{180}=\frac{900 \pi}{180}=\frac{10 \pi}{2}=5 \pi \neq \frac{125 \pi}{9}$
From option (b), $\theta=45^{\circ}$
$\therefore \quad 45^{\circ}=\frac{45^{\circ} \pi}{180}$ radian
Now according to question,
$\frac{45^{\circ} \times 180}{45^{\circ} \pi}=\frac{180}{\pi}$
Now number of degree in $\theta$ is multiplied by number of $\operatorname{radian}$ in $\theta$.
$\therefore \quad 45^{\circ} \times \frac{45^{\circ} \pi}{180^{\circ}}=\frac{45 \pi}{4} \neq \frac{125 \pi}{9}$
From option (c), $\theta=50^{\circ}$
As we know that $180^{\circ}=\pi$ radian
$\therefore \quad 50^{\circ}=\frac{50 \pi}{180}$ radian
Now according to question
$\frac{50^{\circ} \times 180^{\circ}}{50^{\circ} \pi}=\frac{180}{\pi}$
Now number of degree in ' $\theta$ ' is multiplied bynumber of $\operatorname{radian} \operatorname{in} \theta .$
$\therefore \quad 50^{\circ} \times \frac{50 \pi}{180}=\frac{2500 \pi}{180}=\frac{125 \pi}{9}$
$\therefore \quad$ Option (c) is correct.
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