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if $a$ $\pi$ radians is equal to $1440$ degrees, what is the value of $a$ ? (The number of degrees of arc in a circle is $360$ . The number of radians of arc in a circle is $2\pi$ .)

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Coterminal and reference angles

Full solution

Q. if

a

\pi

radians is equal to

1440

degrees, what is the value of

a

? (The number of degrees of arc in a circle is

360

. The number of radians of arc in a circle is

2\pi

.)

Convert Radians to Degrees: We know that $2\pi$ radians is equal to $360$ degrees. Therefore, $a\pi$ radians would be half of $360$ degrees, which is $180$ degrees. However, we are given that $a\pi$ radians is equal to $1440$ degrees. We can set up a proportion to find the value of $a$ .
Set Up Proportion: The proportion is $\frac{a \pi \text{ radians}}{1440 \text{ degrees}} = \frac{1 \pi \text{ radians}}{180 \text{ degrees}}$ . We can solve for $a$ by cross-multiplying.
Cross-Multiply: Cross-multiplying gives us $a \times 180 \text{ degrees} = 1440 \text{ degrees} \times 1$ . Now we can solve for $a$ by dividing both sides of the equation by $180 \text{ degrees}$ .
Solve for $a$ : Dividing both sides by $180$ degrees, we get $a = \frac{1440 \text{ degrees}}{180 \text{ degrees}}$ .
Calculate Final Value: Calculating the division, we find that $a = 8$ . This means that $a \pi$ radians is $8$ times larger than the standard conversion of $\pi$ radians to degrees.

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