Question

1. convert 90° into radians.
2. convert (π/3) radians into degrees.
3. how many radians are there in a full circle?
4. a central angle measures 120° in a circle with radius 10 cm. find the arc length.
5. a circle has a radius of 14 m. if the arc length is 22 m, what is the central angle in radians?
6. find the arc length intercepted by a central angle of (π/6) radians in a circle of radius 12 cm.
7. a wheel of radius 30 cm makes one complete revolution. how far does a point on its rim travel?
8. the arc length of a circle is 15 cm, and the central angle is 60°. find the radius of the circle.
9. a pendulum swings through an angle of 0.8 radians. if its string is 2 m long, find the distance traveled along the arc.
10. a circular track has a radius of 50 m. a runner runs along the track subtending a central angle of 135°. find the distance covered along the arc.

Explanation:

Step1: Recall degree - radian conversion formula

To convert degrees to radians, use the formula $\text{Radians}=\text{Degrees}\times\frac{\pi}{180}$. For $90^{\circ}$, we have $90\times\frac{\pi}{180}=\frac{\pi}{2}$ radians.

Step2: Convert radians to degrees

To convert radians to degrees, use the formula $\text{Degrees}=\text{Radians}\times\frac{180}{\pi}$. For $\frac{\pi}{3}$ radians, we have $\frac{\pi}{3}\times\frac{180}{\pi}=60^{\circ}$.

Step3: Radians in a full - circle

A full - circle is $360^{\circ}$. Using the degree - radian conversion formula, $360\times\frac{\pi}{180} = 2\pi$ radians.

Step4: Find arc length with degree measure

First convert the angle from degrees to radians. $120^{\circ}=120\times\frac{\pi}{180}=\frac{2\pi}{3}$ radians. The arc - length formula is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Given $r = 10$ cm and $\theta=\frac{2\pi}{3}$, then $s=10\times\frac{2\pi}{3}=\frac{20\pi}{3}\approx20.94$ cm.

Step5: Find central angle in radians

The arc - length formula is $s = r\theta$. Given $s = 22$ m and $r = 14$ m, then $\theta=\frac{s}{r}=\frac{22}{14}=\frac{11}{7}\approx1.57$ radians.

Step6: Find arc length with radian measure

Using the arc - length formula $s = r\theta$, with $r = 12$ cm and $\theta=\frac{\pi}{6}$, then $s=12\times\frac{\pi}{6}=2\pi\approx6.28$ cm.

Step7: Distance traveled in one revolution

One complete revolution is $2\pi$ radians. Using the arc - length formula $s = r\theta$, with $r = 30$ cm and $\theta = 2\pi$, then $s=30\times2\pi = 60\pi\approx188.5$ cm.

Step8: Find radius

First convert the angle from degrees to radians. $60^{\circ}=60\times\frac{\pi}{180}=\frac{\pi}{3}$ radians. Using the arc - length formula $s = r\theta$, and given $s = 15$ cm and $\theta=\frac{\pi}{3}$, we can solve for $r$. Rearranging the formula gives $r=\frac{s}{\theta}=\frac{15}{\frac{\pi}{3}}=\frac{45}{\pi}\approx14.32$ cm.

Step9: Distance traveled along the arc

Using the arc - length formula $s = r\theta$, with $r = 2$ m and $\theta = 0.8$ radians, then $s=2\times0.8 = 1.6$ m.

Step10: Find distance covered along the arc

First convert the angle from degrees to radians. $135^{\circ}=135\times\frac{\pi}{180}=\frac{3\pi}{4}$ radians. Using the arc - length formula $s = r\theta$, with $r = 50$ m and $\theta=\frac{3\pi}{4}$, then $s=50\times\frac{3\pi}{4}=\frac{75\pi}{2}\approx117.81$ m.

Answer:

1. $\frac{\pi}{2}$ radians
2. $60^{\circ}$
3. $2\pi$ radians
4. $\frac{20\pi}{3}$ cm
5. $\frac{11}{7}$ radians
6. $2\pi$ cm
7. $60\pi$ cm
8. $\frac{45}{\pi}$ cm
9. $1.6$ m
10. $\frac{75\pi}{2}$ m