Answer the following. Round your answers to the nearest hundredth.
(a) Convert [tex]$\frac{9 \pi}{13}$[/tex] radians to degree measure.
[tex]$\square$[/tex] [tex]$^{\circ}$[/tex]
(b) Convert [tex]$302^{\circ}$[/tex] to radian measure.
[tex]$\square$[/tex] radians
Answer (1)
Convert radians to degrees by multiplying by π 180 . 13 9 π × π 180 = 124.6 2 ∘ .
Convert degrees to radians by multiplying by 180 π . 302 × 180 π = 5.27 radians.
Round both results to the nearest hundredth.
State the final answers: 124.6 2 ∘ and 5.27 radians .
Explanation
Problem Analysis Let's convert radians to degrees and degrees to radians. We'll round our answers to the nearest hundredth as requested.
Converting Radians to Degrees To convert radians to degrees, we use the conversion factor π radians 18 0 ∘ . So, we have: 13 9 π radians × π radians 18 0 ∘ = 13 9 × 180 degrees Calculating this gives us approximately 124.61538... degrees. Rounding to the nearest hundredth, we get 124.6 2 ∘ .
Converting Degrees to Radians To convert degrees to radians, we use the conversion factor 18 0 ∘ π radians . So, we have: 30 2 ∘ × 18 0 ∘ π radians = 180 302 π radians = 90 151 π radians Calculating this gives us approximately 5.27278... radians. Rounding to the nearest hundredth, we get 5.27 radians.
Final Answer Therefore, 13 9 π radians is approximately 124.6 2 ∘ , and 30 2 ∘ is approximately 5.27 radians.
Examples
Understanding how to convert between radians and degrees is crucial in many fields, such as physics and engineering. For example, when designing a satellite dish, engineers need to accurately calculate angles in both radians and degrees to ensure the dish is correctly pointed to receive signals from a satellite. Similarly, in physics, when studying rotational motion, radians are often used because they simplify many formulas. Knowing these conversions allows for precise calculations and accurate designs.
