Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Round the answer to the closest integer.
In 2007, a satellite named SELENE was launched to explore the Moon's surface for a few years. It orbited the Moon in a circular path staying at a constant altitude of kilometers. Given that the radius of the Moon is kilometers, find the distance that the satellite traveled when it completed of its orbit.
radis seldom written. Instead, no unit marker indicates radians. Consider two expressions. The first angle is given in degrees, and the other is given in radians. At first glance, radians might seem inconvenient, but they make calculations simpler in certain circumstances,. Radians are also the SI unit for angles.
Since the measure of a semicircle is equal to radians or the following relation holds true. Using this equality, the conversion factors from degrees to radians and from radians to degrees can be derived.
|Degrees to Radians||Radians to Degrees|
Based on the diagram above, the radian measure of the central angle is defined as follows.
By using this definition, the formula for the length of an arc can be derived.
Here, is the angle measure given in radians. However, what if the measure of a central angle is given in degrees? In that case, by multiplying by the conversion factor the measure can be converted into radians and substituted into the formula. This formula is often written in the following equivalent manner.
This equivalent form is convenient to work with because is the circumference of a circle. Since a full circle measures dividing by results in the length of an arc intercepted by a degree central angle. Finally, by multiplying that value by the measure of the arc, its length is obtained.
In her room Paulina has a big clock whose longer hand is 9 centimeters long. When Paulina came home and looked at her clock, it was exactly She calculated the angle between the longer and shorter hands of the clock to be
To find the length of an arc using the measure of the corresponding central angle given in degrees, the following formula can be used. It is known that the radius of the clock is about centimeters and the central angle that intercepts the arc measures
Finally, Paulina finished all her homework. She could now go to an amusement park with her friends. They get on a Ferris Wheel feeling super excited. The diagram indicates their positions.
It is given that the length of an arc between each cabin is meters. There is one cabin between Paulina and Tiffaniqua, so the arc between their cabins is meters long.
This angle, labeled as is an inscribed angle that intercepts the arc between Paulina and Tiffaniqua. The same arc is intercepted by the central angle found earlier. Therefore, the measure of is half the measure of that central angle. The measure of the angle formed by Paulina, Ali, and Tiffaniqua is about
The challenge presented at the beginning of this lesson can now be solved thanks to the concepts covered.In 2007, a satellite named SELENE was launched to explore the Moon's surface for a few years. It orbited the Moon in a circular path staying at a constant altitude of kilometers. Given that the radius of the Moon is kilometers, find the distance that the satellite traveled when it completed of its orbit.