Thursday, March 08, 2012

Trigonometry: Angles and Arcs, Part 2

CLICK HERE to view the YouTube video that corresponds to this post.

This is the second part of a set of posts dealing with angles and arcs in Trigonometry. In the first part, I covered some definitions pertaining to angles, how to find complements and supplements of given angles, how to find angles coterminal to a given angle, and how to convert between DMS notation and decimal degrees. If you haven't read that post, I highly suggest you do, and you can find it by following this link.

In this part, I will cover arcs and angles measured in radians. We will look at the formula for calculating arc length, as well as how to convert between radians and degrees. At the end, I will give you steps for solving application problems.

First, let's answer a question that I am asked frequently: "What is a radian?" This stumps many students because, prior to taking Trigonometry, they probably never heard the word.

A radian is the measure of an angle formed when an arc of length r is subtended on a circle of radius r. This means that if you have a circle of any radius and you measure an arc length equal to that radius, then the angle formed by that arc length is defined as exactly 1 radian.

There are π (pi) radians in a half circle and 2π radians in a full circle. Therefore, π radians correspond to 180 degrees and 2 π radians correspond to 360 degrees.

An important element of Trigonometry is converting between radians and degrees. These are the two most commonly used units of measure for angles. All you need to remember is the conversion factor I just gave you (and also shown in the blue box below).

How to Convert Between Radians and Degrees
Let's try some example problems. In the problems below, I've boxed the angle we are converting in blue, the conversion factor we are using in green, and the answers in red. After multiplying by the correct conversion factor, you simply need to reduce fractions and simplify as much as possible. A common error students make is to use 2 π radians instead of just π radians in the conversion factor; avoid this by memorizing the conversion factor properly.

You can also use your calculator to convert between radians and degrees, though it is not very efficient to do so. In the beginning, you can use this method to check your answers for converting between degrees and radians.

How to Use Your Calculator to Convert Between Degrees and Radians

Step 1: Make sure your calculator is in the mode corresponding to the unit you are converting TO. To check what mode your calculator is in, press the MODE key (next to the 2ND key). In this menu, about 3 options down, you will see RADIAN DEGREE, with one of these highlighted (look at red box in images below). If the one highlighted is not the unit you are converting TO, arrow down to this row and then arrow over and press ENTER on the unit you want to choose. Below you can see the menu options. In my MODE menu on the left, RADIAN is highlighted, which indicates that my calculator is in RADIAN mode. I can only convert FROM degrees TO radians in this mode. If I want to convert FROM radians TO degrees, I have to change it to degree mode, as shown on the right.
Step 2: Press 2ND then MODE to exit the MODE menu. Input the angle you wish to convert. To input DEGREES, press 2ND then APPS to enter the ANGLE menu. Choose option 1 for the degree symbol (boxed in red below). To input RADIANS, press 2ND then APPS to enter the ANGLE menu and choose option 3 for the radians symbol (boxed in blue below).
Step 3: Press ENTER to tell your calculator to make the conversion.

There are two things you need to note when using your calculator to make this conversion.

1. For fractional radian measures, you MUST input the fraction in parentheses before inserting the radians symbol. In the image below, you can see that if I do not put parentheses around the fraction, the computer does not arrive at the correct answer! (See below)

2. When converting TO radians, the calculator will not show you the answer in terms of π. It will, instead, give you a decimal approximation. Do not use this decimal as your answer. Instead, convert it back into a fraction. First, divide by π and convert into a fraction (press MATH and choose 1: >FRAC to have it convert the fraction for you). Then simply multiply by π in the numerator to get your final converted answer.
I hope by now you can see that using the calculator to work through conversions is definitely not as quick as simply multiplying by the conversion factor!

Next, let's look at arc length. Arc length is the linear distance of a circle. Imagine a very large circle on the ground. If you were to walk along the circumference of this circle, you'd be walking an arc. To calculate this linear arc length, you use the arc length formula. Note that the angle measure MUST be in radians when using this formula. If the angle is not given in radians, you must convert it before using the angle measure.

Here is an example problem for finding the arc length.
Similarly, you can work problems with other variables given. If you are given arc length and radius, you can solve for the angle. If you are given arc length and angle, you can solve for the radius of the circle.

Application problems dealing with arc length often confuse students, but they are not difficult to work. Pulleys are often the stage for arc length questions. Let's work an example together.

How to Solve Arc Length Application Problems
Step 1: Draw a picture to represent the situation at hand, if one is not given already. Consider my drawing in the example. I have a larger pulley connected to a smaller pulley. I identified the larger pulley as the first pulley. Use subscripts to distinguish between the parts of one pulley versus another.

Step 2: Label all variables on the image, and then list all known variables. In my example, I am given the radius of each pulley and the angle measure of the larger pulley. The question is asking what angle the smaller pulley makes when the larger makes one revolution. Remember that in the arc length formula, the angle must be in radians, so I have quickly converted 1 revolution into radians. One revolution is equal to 2π radians.

Step 3: Set up equations for the variables you've labeled. In the example, I've written the equations for the arc lengths of each individual pulley.

Step 4: Make a connection between equations if you have more than one. Here is where students get thrown off. When two pulleys are connected by a belt, the arc length that one pulley describes is the same as the arc length the other pulley describes. The difference lies in the angles they create. The smaller pulley will subtend the larger angle. In order for the belt to move the same distance along the smaller pulley, it has to make a larger angle because the pulley is smaller. Think of this: there are two bicycles. One has tires with smaller diameters than the other bicycle. If you were going to ride one a long distance, which would you choose? You should choose the one with larger tires. Larger tires mean you do less pedaling because in an equal number of revolutions (pedals), a larger tire is going to cover more ground (it has a larger circumference). The pulleys work similarly. So set up the arc lengths equal to each other and plug in the equations.

Step 5: Before plugging in known values, solve for the unknown variable. Then, plug in known values and perform all necessary calculations. Here, you want to leave radians in terms of π. The question asks you to give the answer in terms of radians and degrees. After finding the answer in terms of radians, convert your answer into degrees.
This concludes our look at angles and arcs.