Radian and Degrees Lesson Plan (Part 1) By: Douglas A. Ruby Class: Pre-Calculus II

Date: 10/10/2002 Grades: 11/12

INSTRUCTIONAL OBJECTIVES: At the end of this two-part lesson, the student will be able to: 1. Given an angle in decimal degrees form, be able to convert to degrees, minutes, and seconds. 2. Given an angle in degrees, minutes, and seconds, convert to decimal degrees. 3. Given a positive3 or negative angle in degree measure, convert to radian measure for angles of /6, /4, /3, or /2 radians or equivalent in any of the 4 quadrants without using a calculator. 4. Given a positive or negative angle in radian measure, convert to degree measure for angles of 30o, 45o, 60o, or 45o degrees or equivalent in any of the 4 quadrants without using a calculator. 5. Convert between degree and radian measurement for arbitrary positive or negative angles when using a calculator 6. Reproduce all six trigonometric functions for a given angle of /6, /4, /3, or /2 or its equivalent in any of the 4 quadrants without using a calculator 7. Reproduce all six trigonometric functions for any given positive or negative angle in any of the 4 quadrants using a calculator Relevant Massachusetts Curriculum Framework PC.M.1 – Describe the relationship between degree and radian measures, and use use radian measure in the solution of problems, particularly problems involving angular velocity and acceleration. PC.P.3 - Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions. MENTAL MATH – (5 Minutes)

Question 1: Provide one positive and one negative angle are co-terminal with: a) 24o 384o -336o b) 48.83 408.83o -311.17o o o c) –423.93 -63.93 296.07o Note that the co-terminal angle when you rotate from a positive decimal to a negative decimal changes! For instance, -311.17o is co-terminal with +48.84o. The decimal part is not the same. Question 2: Recall the discussion yesterday on positive and negative angles. What is the sine of the following angles? (use your calculator) a) 59.73o .8637 b) 135o -.707 or –sqrt(2)/2 o c) -585 -.707 or –sqrt(2)/2 Page 1

Degrees and Radians – Mr. Ruby CLASS ACTIVITIES – (Note: 45 Minute Lesson Plan) Warm-Up after Mental Math: Let’s start today’s discussion with the diagram to the left. If we II remember yesterday’s discussion, what are the 45 “standard” 30o, 45o, and 60o angles in each quadrant? Yes, in Q I, we have 30o, 45o, and 60o. In Q II we have 30 120o, 135o, and150o. In Q III, we have 210 o, 225o, and o o o o 240 . In Q IV, we have 300 , 315 , 330 . And what = 135o about the angles that are described by the terminal x side being on one of the four points of the x/y axes? Yes, those are 0o, 90o, 180o, 270o, and 360o. Who can give me a positive angle that is co-terminal with III IV 135o? An angle such as 495o would be coterminal. What about a negative angle that is coterminal with 135o? An angle such as -225o or –585o would be coterminal. Can you come up with an equation that describes any angle that is coterminal with 135o? Any angle satisfying the equation = (135o+ n 360o) where n is ANY integer would be coterminal. Good! y

o

60

I

o

o

1. Conversion between Decimal Degrees and Minutes/Seconds Ok, pull out your calculators. So far, with just a few exceptions, we have limited our discussion of degree measurement to examples containing only decimal degrees of the form: = 59.73o (read as “fifty-nine point seven degrees”) Now, whether we like it or not, we have a nearly 5000 year heritage, handed to us by the ancient Mesopotamians and Babylonians. They had a Base 60 number system. Do you know how their Base60 number system affects us today? Yes, it is the way we tell time with the clock (minutes and seconds) and the way we measure angles and compass headings with degrees, minutes, and seconds. So we need to be able to convert between decimal degree measurements and measurements in the form of: = 59o 43’ 48” (read as “fifty-nine degrees, forty-three minutes, forty-eight seconds”) So what do you suppose an angle of 1 minute is equal to? One sixtieth of a degree. And what is an angle of one second equal to? One sixtieth of a minute or 1/3600th of a degree. In our prior example, we had a degree measurement of 59.73o. We can leave the degree portion alone and just convert the decimal portion of the measurement to minutes and seconds, since we know that after conversion, we still have 59-and-a-few degrees. I use the remainder method with my calculator to convert from decimal to minutes/second degree measurement. It goes like this: Follow with me on your calculators. Take the .73o that remains after the decimal point and multiply by 60 (because there are 60 minutes in a degree). What do you get? You get .73o = Page 2

Degrees and Radians – Mr. Ruby 43.8’. Ok, so now we know that degree measurement of 59.73o = 59o 43.8’. But we still have a decimal (non-integer) number of minutes. How do we convert that to seconds? RIGHT! We take the decimal remainder of .8 and multiply by 60 because there are 60 seconds in a minute of rotation. What do we get? You get 0.8’ = 48” So now that we know that 43.8’ equals 43’ 48”, we have completed our conversion by re-writing: = 59.73o = 59o 43’ 48” So to summarize: To convert from decimal degrees to minutes/seconds we: a) Keep the integer portion of the degree measurement b) Multiply the decimal remainder from a) by 60. Use the integer portion of the result as the number of minutes. c) Multiply the decimal remainder from b) by 60. This result is the number of seconds. Round this number to the nearest second. Let’s try a few conversions: Convert the following angles to minutes and seconds to the nearest second: a)

= 67.31o 67o 18’ 36”

b)

= 47.38o 47o 22’ 48”

c)

= 85.06312o 85o 3’ 47”

Notice on the last example that the number of seconds remaining after we converted to minutes was .7872. This converts to 47.232”. But we rounded it to the nearest second to get 85o 3’ 47”. 2. Conversion between Minutes/Seconds and Decimal Degrees Now we want to go the other way. We want to convert from degrees with minutes and seconds to decimal degrees. This will become important as we convert to and from radian measurement. Lets think about what kind of equation can be used to convert from an angle like: -134o 47’ 22” …to its decimal equivalent. Given what we just have done, how would we do this conversion? Well, it actually is fairly easy. If you have a graphing calculator, you can just punch in a single equation. If you have a standard calculator, you will do this piece-wise, but the approach is as follows: decimal degrees = integer degrees + (1/60

minutes) + (1/3600

So in our example above, we get: -134o 47’ 22” = -134

1 47 60

1 22 = 3600

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134.7894 o

seconds)

Degrees and Radians – Mr. Ruby where the “4” at the end repeats. Isn’t that easy? Let’s try a few conversions: Convert the following angles to decimal degrees from minutes and seconds. Roudn to the nearest 1/1000 of a degree: a)

= 12o 03’ 14” o 12.0539

b)

= 73o 45’ 11” o 73.7531

c)

= 230o 53’ o 230.8833

Great! Now we know how to handle all examples of decimal and minute/second degree measurements. Lets wrap up part A of our Lesson with some examples of trigonometric functions with angles in minutes and seconds. 3. Trigonometric Functions of angles in Minutes/Seconds and Decimal Degrees Now that we understand all of the conversions from decimal degrees to minutes/seconds and back, we are going to use that knowledge to be able to use our calculators to find the trigonometric function value for any arbitrary angle. To do this, lets use the following example: Example: Calculate the tangent of the angle 1322o 52’ 47” Now, we can do the following with our TI-83+ Calculators. Make sure you have your calculator in Degree mode (see below). Our calculators want the argument to the tangent function to be a decimal number (not in degrees and minutes), so we need to convert the angle to decimal form. We can do this as part of entering our tangent function. As you can see in the example to the right, the tangent of 1322o 52’ 47” is 1.95246… So to summarize, to find the trigonometric function of an angle expressed in degrees, minutes, and seconds, we need to: a. Make sure our calculators are in degree mode. b. Convert the degree measurement to decimal degree measurement c. Find the trigonometric function value for the decimal degree measurement. Here are a few examples to try using the angles from above: a) sin 12o 03’ 14” sin 12.0539o=.2088

b) cot 73o 45’ 11” cot 73.7531o= .2539

c) sec 230o 53’ sec 230.8833o=-1.585

4. Homework Review (Part 1) Great, now let’s review the homework. You have 12 problems to do. In each, convert the angle to decimal form and find the trigonometric function value indicated. In addition, tell me which quadrant the angle is in.

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Degrees and Radians – Mr. Ruby

HOMEWORK: Source: -

Bittinger and Beecher, Algebra and Trigonometry, Section 6.4, pp. 391-404 Joyce, David E., Dave's Short Trig Course, Sections 2, 9, and 10 http://aleph0.clarku.edu/~djoyce/java/trig/angle.html

Homework Part 1 Find the following function values. In which quadrant is each angle? For the angles that are more negative than -360o or more positive than 360o, what positive or negative angle (such that meets the requirement that 359o 59’ 59”” < < 359o 59’ 59”) is each co-terminal with? Example: cot –422o 37’ cot –422o 37’ = 1/tan(-422-37/60) = -.5182 This angle is in Quadrant IV and is co-terminal with the angle –62o 37’ Function 1. tan 295° 14'

Value -2.122

Quadrant QIV

2. cos 230o 53'

-0.6309

QIII

3. sec 146.9°

-1.194

QII

4. sin 98.4°

0.9893

QII

5. sin 756° 25'

0.5937

QI

36o 25’

6. cot 820° 40'

-0.1883

QII

100o 40’

7. cos (-1000.85°)

0.1882

Q1

-280.85°

8. tan (-1086.15°)

-0.1078

QIV

-6.15o

9. cot (-16° 37')

-3.351

QIV

10. csc (-13°28')

-4.294

QIV

11. sin 3824°

-0.6947

QIII

224o

12. cos 5417°

0.9563

QI

17o

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Co-terminal Angle

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