5. convert the following degrees into their positive radian measures between 0π and 2π.\na. 54° b. 75° c…
5. convert the following degrees into their positive radian measures between 0π and 2π.\na. 54° b. 75° c. -180° d. 227° e. -325° f. 207°
Answer
Explanation:
Step1: Use the conversion formula
To convert degrees to radians, use the formula ( \text{Radians}=\frac{\pi}{180}\times\text{Degrees} ).
Step2: Calculate for each part
- a. For (54^{\circ}) [ \begin{align*} \text{Radians}&=\frac{\pi}{180}\times54\ &=\frac{3\pi}{10} \end{align*} ]
- b. For (75^{\circ}) [ \begin{align*} \text{Radians}&=\frac{\pi}{180}\times75\ &=\frac{5\pi}{12} \end{align*} ]
- c. For (- 180^{\circ}) [ \begin{align*} \text{Radians}&=\frac{\pi}{180}\times(-180)\ &=-\pi \end{align*} ] Since we need positive radian measures between (0\pi) and (2\pi), (-\pi + 2\pi=\pi)
- d. For (227^{\circ}) [ \begin{align*} \text{Radians}&=\frac{\pi}{180}\times227\ &=\frac{227\pi}{180} \end{align*} ]
- e. For (-325^{\circ}) [ \begin{align*} \text{Radians}&=\frac{\pi}{180}\times(-325)\ &=-\frac{65\pi}{36} \end{align*} ] (-\frac{65\pi}{36}+2\pi=-\frac{65\pi}{36}+\frac{72\pi}{36}=\frac{7\pi}{36})
- f. For (207^{\circ}) [ \begin{align*} \text{Radians}&=\frac{\pi}{180}\times207\ &=\frac{23\pi}{20} \end{align*} ]
Answer:
a. (\frac{3\pi}{10}) b. (\frac{5\pi}{12}) c. (\pi) d. (\frac{227\pi}{180}) e. (\frac{7\pi}{36}) f. (\frac{23\pi}{20})