Time For Trigonometry!!! Degrees, Radians, Revolutions Arc Length, Area of Sectors SOHCAHTOA Unit Circle Graphs of Sine, Cosine, Tangent Law of Cosines Law of Sines Lesson 5 1 Objectives Convert between degrees, radians & revolutions. Memorize angles in radians (multiples of 30 and 45 degrees).
If I asked you if you remember doing some work with radians in Advanced Algebra, you would probably say yes, but do you think you could define what a radian is? What is the following fraction equivalent to? 180 o π radians If you said 1, then you are right! Just like inches and centimeters have conversions for distance, degrees and radians have conversions for angle measures. It just so happens that 180 degrees is equivalent to pi radians.
Lesson 5 1: Measures of Angles We are going to look at angles measured in degrees, revolutions, or radians. It will be important that you can convert between these three forms because throughout the chapter you will find it more useful to use one form over another. The following are equivalent to each other: 360 degrees = 1 revolution = 2 π radians r r A radian is defined to be the measure of the angle formed when the arc length and radius of a circle are the same. Notice the two red lengths are both of length r in the picture. 1 radian = π/180 57.3 degrees Why do we use radians? Because there are times where degrees don't make much sense or are a pain to work with. The motion of a Ferris wheel, the ebb and flow of the tide, and many other situations can be modeled using sine and cosine functions and it usually is easier to utilize radians rather than degrees in these situations. As you will see in this lesson, radians can be easier to convert without a calcualtor then degrees.
Example 1 Convert 3/5 revolutions counterclockwise into degrees and radians. There are two ways to convert angles to degrees and radians. You can set up and solve a proportion (method #1) or you can multiply the revolutions by the degrees or radians in a complete revolution (method #2 on next slide) Notice how much easier the radian conversion is to do without the aid of a calculator!!! Example 1 Convert 3/5 revolutions counterclockwise into degrees and radians.
What would be different if example one had said clockwise instead? The clockwise direction represents negative angles. This would mean the answers would look as follows: So 3/5 of a revolution clockwise is equivalent to 216 degrees or 6π/5 radians. Example 2 Convert 100 o into radians and revolutions. You will constantly be asked to convert between degrees and radians throughout this chapter. Therefore, this problem is vital!!! You still have two methods for this conversion, a proportion or a multiplication rule.
Example 2 Convert 100 o into radians and revolutions. Example 3 Convert 17π/10 into degrees and revolutions. You will constantly be asked to convert between degrees and radians throughout this chapter. Therefore, this problem is vital!!! You still have two methods for this conversion, a proportion or a multiplication rule.
Example 3 Convert 17π/10 into degrees and revolutions. Convert each degree measure into radians. 150 0 135 0 120 0 90 0 60 0 45 0 30 0 Do not use a calculator. All of these angle measures need to be memorized!!! 180 0 360 0 210 0 330 0 225 0 315 0 240 0 270 0 300 0 If you can figure out 30 degrees and 45 degrees, you should be able to use those to determine all of the rest. The following slides show how to use 30 degrees and 45 degrees to complete all of the conversions with minimal hassle.
Convert each degree measure into radians. 135 0 120 0 90 0 60 0 45 0 Do not use a calculator. All of these angle measures need to be memorized!!! 150 0 30 0 180 0 360 0 210 0 330 0 225 0 315 0 240 0 270 0 300 0 180 o is equivalent to π radians. There are six 30 o increments in 180 o, so 30 o = π/6 radians. Notice the only values I am missing are the ones circled in red!!! Convert each degree measure into radians. 150 0 135 0 120 0 90 0 60 0 45 0 30 0 Do not use a calculator. All of these angle measures need to be memorized!!! 180 0 360 0 210 0 330 0 225 0 315 0 240 0 270 0 300 0 180 o is equivalent to π radians. There are four 45 o increments in 180 o, so 45 o = π/4 radians. Notice the only values I am missing are circled in blue (but the 30 o already took care of those)!!!
Homework Lesson Master 5 1 & Degrees Radians Remember the Chapter 1 3 Review Worksheet is being collected tomorrow. If you find yourself having trouble in this chapter or previous ones, come see me and let's get you caught up before the final. It is 31 school days away!!!