Convert [tex]$450^{\circ}$[/tex] to radian measure in terms of [tex]$\pi$[/tex].
[tex]$\square$[/tex] radians
Convert [tex]$-\frac{15 \pi}{4}$[/tex] radians to degree measure.
[tex]$\square$[/tex]
Answer (1)
Convert degrees to radians by multiplying by 180 π . 45 0 ∘ × 180 π = 2 5 π .
Convert radians to degrees by multiplying by π 180 . − 4 15 π × π 180 = − 67 5 ∘ .
The radian measure of 45 0 ∘ is 2 5 π .
The degree measure of − 4 15 π is − 67 5 ∘ .
Explanation
Problem analysis We are given two conversion problems: (a) Convert 45 0 ∘ to radian measure in terms of π .
(b) Convert − 4 15 π radians to degree measure.
Converting degrees to radians (a) To convert degrees to radians, we multiply the degree measure by 180 π . Therefore, we have: 45 0 ∘ × 180 π = 180 450 π We simplify the fraction 180 450 . From the previous calculation, we know that 180 450 = 2.5 = 2 5 .
So, 45 0 ∘ = 2 5 π radians.
Converting radians to degrees (b) To convert radians to degrees, we multiply the radian measure by π 180 . Therefore, we have: − 4 15 π × π 180 = − 4 15 × 180 We simplify the fraction 4 15 × 180 . From the previous calculation, we know that − 4 15 × 180 = − 675 .
So, − 4 15 π = − 67 5 ∘ .
Final Answer (a) 45 0 ∘ is equal to 2 5 π radians. (b) − 4 15 π radians is equal to − 67 5 ∘ .
Examples
Understanding radian and degree conversions is crucial in fields like astronomy and engineering. For instance, astronomers use radians to measure angles in the sky when tracking celestial objects, while engineers use them in calculations involving circular motion and waveforms. Knowing how to convert between these units allows for accurate measurements and calculations in various scientific and technical applications.
