Applications of radian measure: Practical examples in Pre-Calculus

📚 Quick Study Guide

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• Radian Definition: A radian is the measure of an angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
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• Conversion: To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$.
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• Arc Length: The length $s$ of an arc subtended by a central angle $\theta$ (in radians) in a circle of radius $r$ is given by $s = r\theta$.
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• Area of a Sector: The area $A$ of a sector with central angle $\theta$ (in radians) and radius $r$ is given by $A = \frac{1}{2}r^2\theta$.
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• Angular Speed: If an object moves along a circular path of radius $r$ with angular speed $\omega$ (in radians per unit time), then its linear speed $v$ is given by $v = r\omega$.

Practice Quiz

1. Question 1: A circle has a radius of 6 cm. Find the arc length subtended by a central angle of $\frac{\pi}{3}$ radians.
   1. 2$\pi$ cm
   2. 3$\pi$ cm
   3. 4$\pi$ cm
   4. $\pi$ cm
2. Question 2: A sector of a circle has a radius of 4 inches and a central angle of $\frac{3\pi}{4}$ radians. What is the area of the sector?
   1. 4$\pi$ square inches
   2. 5$\pi$ square inches
   3. 6$\pi$ square inches
   4. 3$\pi$ square inches
3. Question 3: A wheel with a diameter of 1 meter is rotating at a rate of 60 revolutions per minute. What is the angular speed of the wheel in radians per second?
   1. $\pi$ rad/s
   2. 2$\pi$ rad/s
   3. 3$\pi$ rad/s
   4. 4$\pi$ rad/s
4. Question 4: A pendulum swings through an angle of 10 degrees. If the pendulum is 2 meters long, how far does the tip of the pendulum travel?
   1. $\frac{\pi}{9}$ meters
   2. $\frac{\pi}{18}$ meters
   3. $\frac{\pi}{36}$ meters
   4. $\frac{2\pi}{9}$ meters
5. Question 5: Convert 225 degrees to radians.
   1. $\frac{3\pi}{4}$
   2. $\frac{5\pi}{4}$
   3. $\frac{7\pi}{4}$
   4. $\frac{\pi}{4}$
6. Question 6: A bicycle wheel has a radius of 30 cm. If the bicycle travels 10 meters, through what angle (in radians) does the wheel rotate?
   1. $\frac{100}{3}$
   2. $\frac{10}{3}$
   3. $\frac{1}{3}$
   4. $\frac{3}{100}$
7. Question 7: Find the area of a sector of a circle with radius 5 cm and arc length 7 cm.
   1. 17.5 $cm^2$
   2. 35 $cm^2$
   3. 10 $cm^2$
   4. 70 $cm^2$