Problem 82
Question
In this set of exercises, you will use degree and radian measure to study real-world problems. What is the angle swept out by the second hand of a clock in a 20 -second interval? Express your answer in both degrees and radians.
Step-by-Step Solution
Verified Answer
The angle swept out by the second hand of a clock in a 20-second interval is 120 degrees or \(\frac{2\pi}{3}\) radians.
1Step 1: Calculate the proportion of time
Divide the given time interval (20 seconds) by the total time taken for the second hand of a clock to complete a full circle (60 seconds). Therefore, the proportion of time is \(\frac{20}{60} = \frac{1}{3}\)
2Step 2: Convert the proportion to degrees
Multiply the proportion calculated in step 1 by the total degrees in a circle (360 degrees). Therefore, the angle in degrees is \(\frac{1}{3} * 360 = 120\) degrees.
3Step 3: Convert the proportion to radians
Multiply the proportion calculated in step 1 by the total radians in a circle (\(2\pi\) radians). Therefore, the angle in radians is \(\frac{1}{3} * 2\pi = \frac{2\pi}{3}\) radians.
Key Concepts
Degree and Radian MeasureProportional Reasoning in MathematicsReal-world Applications of Trigonometry
Degree and Radian Measure
Understanding the relationship between degree and radian measure is fundamental in trigonometry and forms the basis of many real-world applications. Degrees and radians are two units for measuring angles, and they're convertible.
One complete revolution around a circle corresponds to 360 degrees, which is equivalent to \(2\pi\) radians. To convert from degrees to radians, we multiply the number of degrees by \(\frac{\pi}{180}\), and to convert from radians to degrees, we multiply the number of radians by \(\frac{180}{\pi}\).
In the context of the exercise, we are asked to find the angle swept by the second hand in a 20-second interval. Clocks are a great example to visualize angles because their faces are essentially 360-degree circles. Given that the second hand makes a full circle, or 360 degrees in 60 seconds, for every second that passes, the hand sweeps \(\frac{360}{60} = 6\) degrees. Therefore, in 20 seconds, it sweeps \(20 \times 6 = 120\) degrees. Using the conversion factor, the radians swept can similarly be calculated, offering an alternative measure of the angle.
One complete revolution around a circle corresponds to 360 degrees, which is equivalent to \(2\pi\) radians. To convert from degrees to radians, we multiply the number of degrees by \(\frac{\pi}{180}\), and to convert from radians to degrees, we multiply the number of radians by \(\frac{180}{\pi}\).
In the context of the exercise, we are asked to find the angle swept by the second hand in a 20-second interval. Clocks are a great example to visualize angles because their faces are essentially 360-degree circles. Given that the second hand makes a full circle, or 360 degrees in 60 seconds, for every second that passes, the hand sweeps \(\frac{360}{60} = 6\) degrees. Therefore, in 20 seconds, it sweeps \(20 \times 6 = 120\) degrees. Using the conversion factor, the radians swept can similarly be calculated, offering an alternative measure of the angle.
Proportional Reasoning in Mathematics
Proportional reasoning is a key component in mathematics, particularly when working with ratios and similar figures. It allows us to solve for unknown quantities by setting up a proportion, which is an equation that equates two ratios.
In the exercise, we use proportional reasoning to relate the 20-second interval to the full 60-second rotation of the clock's second hand. The proportion of time that has passed is \(\frac{20}{60}\), or simply \(\frac{1}{3}\). This principle can be applied to countless scenarios where a part-to-whole relationship needs to be determined. By understanding proportions, we can scale up or scale down quantities and make predictions or conversions in various fields, from cooking to construction, and in this case, time to angle measure.
In the exercise, we use proportional reasoning to relate the 20-second interval to the full 60-second rotation of the clock's second hand. The proportion of time that has passed is \(\frac{20}{60}\), or simply \(\frac{1}{3}\). This principle can be applied to countless scenarios where a part-to-whole relationship needs to be determined. By understanding proportions, we can scale up or scale down quantities and make predictions or conversions in various fields, from cooking to construction, and in this case, time to angle measure.
Real-world Applications of Trigonometry
Trigonometry may seem abstract, but it has many practical applications in the real world. It's utilized in fields such as engineering, physics, architecture, and even in areas like geography and astronomy. The principles of trigonometry help us understand phenomena such as sound waves, light, and the orbits of planets.
For instance, architects use trigonometry to determine the correct slope of a roof, and engineers might use it to calculate the stress on a particular part of a structure. In navigation, the concept of angles and their measurement in degrees or radians is crucial for setting a proper course. The exercise about the clock's second hand, albeit simple, is a practical application. It shows how understanding angles and their measures can help us calculate precise intervals and positions in timekeeping, which is just one small cogwheel in the complex machinery of real-world trigonometry.
For instance, architects use trigonometry to determine the correct slope of a roof, and engineers might use it to calculate the stress on a particular part of a structure. In navigation, the concept of angles and their measurement in degrees or radians is crucial for setting a proper course. The exercise about the clock's second hand, albeit simple, is a practical application. It shows how understanding angles and their measures can help us calculate precise intervals and positions in timekeeping, which is just one small cogwheel in the complex machinery of real-world trigonometry.
Other exercises in this chapter
Problem 81
In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal place
View solution Problem 82
Find the exact value of each expression without using a calculator. $$2 \sin \frac{\pi}{6}-\cos \frac{\pi}{3}$$
View solution Problem 82
In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal place
View solution Problem 83
Find the exact value of each expression without using a calculator. $$\sin \frac{\pi}{4} \cos \frac{\pi}{4}$$
View solution