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Which of the following radian measures is equal to $720^\circ$ ? (The number of degrees of arc in a circle is $360^\circ$ . The number of radians of arc in a circle is $2\pi$ .)

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Coterminal and reference angles

Full solution

Q. Which of the following radian measures is equal to

720^\circ

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

360^\circ

. The number of radians of arc in a circle is

2\pi

.)

Set up proportion: To convert degrees to radians, we use the fact that $360$ degrees is equal to $2\pi$ radians. Therefore, to find the radian measure equivalent to $720$ degrees, we set up a proportion.
Cross-multiply: We know that $360$ degrees $= 2\pi$ radians. So, we can write the proportion as: $\newline$ $(720$ degrees $) / (360$ degrees $) = (x$ radians $) / (2\pi$ radians)
Isolate x: Now we solve for x by cross-multiplying: $720 \text{ degrees} \times 2\pi \text{ radians} = x \text{ radians} \times 360 \text{ degrees}$
Simplify fraction: Divide both sides by $360$ degrees to isolate $x$ : $\frac{720}{360} \times 2\pi = x$
Find $x$ : Simplify the fraction $\frac{720}{360}$ , which equals $2$ : $2 \times 2\pi = x$
Final result: Multiply $2$ by $2\pi$ to find $x$ : $\newline$ $2 \times 2\pi = 4\pi$
Final result: Multiply $2$ by $2\pi$ to find $x$ : $\newline$ $2 \times 2\pi = 4\pi$ So, the radian measure equivalent to $720$ degrees is $4\pi$ radians.

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