CMD204 Lab 6 Radians PDF

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Lab report for lab 6: radians...

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1 CMD 204 Lab Activity 6: Radians: “A slice of Pi”

Name: Ella MacCallum

Equipment: 1. 2. 3. 4. 5. 6.

1 paper plate 1 length of string (more than enough to go around the edge of the plate) Scissors Pencil Ruler Protractor

Background: The radian is an angular measure based on the unit circle.

A radian is the angle subtended at the center of any circle by an arc whose length is equal to the radius of the circle.

Angles are parts of circles. Angles and circles are everywhere in nature, visibly and invisibly. Because of this, radians are widely used in Math, Physics and the applied sciences. Believe it or not, for scientists, radians are simpler to deal with than 360 arbitrary degrees! It was the Babylonians who decided to divide the circle into 360 parts, because they used a base 60 number system (which is also where we got hours, minutes and seconds from). So, you are used to using degrees, but consider this: the deepest and most fundamental workings of Nature, described by trigonometry, calculus, symmetries, and other mysterious patterns that make the universe tick, those workings don’t care about 360 degrees! They are much more involved with π. Although π might seem weird to you, it is very special indeed. π is the ratio of a circle’s circumference to its radius.  

Recall that the circumference of a circle with radius r is 2πr. Recall that a ratio is a relationship between two numbers indicating how many times the first number contains the second.

Using π, radians and ratios lets us explore Nature very deeply as we seek to quantify phenomena such as sound waves on the electromagnetic spectrum. Today we will explore radians visually so that we understand them.

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Procedure: 1. Using the string, carefully measure the circumference of your paper plate. Do this by marking where the end of your string begins on the plate and then carefully work the string around the edge of the plate until you reach your original mark. Record this length: 74.295 cm 2. Fold plate in half. Crease the fold line so that it can be clearly seen. 3. Fold it again into quarters. Crease the fold lines. 4. Open the plate. Using the ruler, draw line segments along the fold lines forming four quadrants. Label the points on the outer edge of the plate that correspond to 0º, 90º, 180º and 270º. 5. The folding process has located the center of your circle. Use your string to measure the radius of the circle. Cut the string to this length. Record this length: 11.43 cm 6. Hold one end of the radius length string at the edge of the plate at 0º. Wrap the string around the edge of the plate and mark its ending location. Connect this point to the center of the circle. 7. Using your protractor, find the number of degrees in the central angle formed from 0º to the segment you drew in step 6. ***In terms of radians, this angle has a measure of one (1) radian*** Record this answer: 55 degrees 8. Using your radius length string, continue to wrap the string around the edge of the plate marking its ending locations. Record the number of radian angles that fit in the circle. 6.25 ***Hint: this will not be a whole number!***

9. Thought exercise: Recall again that the circumference of a circle with radius r is 2πr.

3 Now calculate this: 2π(11.43)= 71.63 Plate circumference / string length = 71.63/11.43= 6.28 What does this number look like? Does it bear any relation to π? _Describe: This number is similar to the amount of radians that we calculated, making 6.28 directly related to 2π. If you multiply pi, or 3.14 by 2, you get 6.28. Therefore, 6.28 does bear a direct relation to pi.

Good to know: Conversion between degrees and radians is easy!...