1 / 10

Some trigonometry problems

Some trigonometry problems. From Fred Greenleaf’s QR text Samuel Marateck. Problem 1. Because angular measure is used to calculate arc length, angles are measured In radians. There are 2 π radians in a circle since the circumference of a unit circle is

bperron
Download Presentation

Some trigonometry problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Some trigonometry problems From Fred Greenleaf’s QR text Samuel Marateck

  2. Problem 1 Because angular measure is used to calculate arc length, angles are measured In radians. There are 2 π radians in a circle since the circumference of a unit circle is 2 π. The arc length s of a sector of a circle of radius r for an angle Ө is s= Ө r.

  3. To convert an angle measured in degrees to radians do the following: Find the fraction of 3600 the angle represents and multiply it by 2 π. So an angle of 300 is 30/360 * 2 π or 0.52 radians.

  4. Eratosthenes measured the circumference of the earth by taking measurements of the angle of the sun to the vertical. The vertical is measured by suspending a weight from a string (it’s called a plumb line).The weight will point to the center of the earth. They took measurements in two places in Egypt 500 miles apart. The first in Alexandria and the second in Syene.

  5. At noon the sun shone perpendicular to a tangent plane on the earth’s surface in Syene. In Alexandria, the sun’s rays made a 7.50 angle with the perpendicular. But this is the angle of a sector made with the radii drawn from these two cities to the earth’s center. So s, which equals 500 miles, = 7.50 R, where R is the earth’s radius.

  6. 500 = 2*3.1459 * 7.5/360 * R So R = 500*360/(2*3.14159 * 7.5) = 3820 miles.

  7. Problem 2 How far is the horizon from you if you are 30 feet (call it b when expressed in miles) above the earth’s surface? Your line of sight is tangent to the earth’s surface at the horizon. Let’s call the distance to the horizon a. A line drawn from the horizon to the earth’s center is perpendicular to your line of sight. We let r be the earth’s radius.

  8. The line from you to the earth’s center has length r + b and forms thehypotenuse of a right triangle with legs r and a. The Pythagorean theorem gives: (r + b)2 = r2 + a2 So r2 + b2 + 2br = r2 + a2 cancel r2 on both sides. Since b is 30ft/5280ft, b2 is very small, so set it to 0.

  9. So r2 + b2 + 2br = r2 + a2 becomes 2br = a2 . So a=√(2br). r = 3958miles and b = 30/5280, so a = √ (2*30*3958/5280) = √ (45) a = 6.7 miles

  10. You may have learned in grammar school that the intelligentsia in Columbus’ time thought that the earth was flat. This was a myth promulgated by Washington Irving in Life and Voyages of Columbus, 1830. It was debunked by Samuel Eliot Morison who described Irving’s description of Columbus as “malicious nonsense” in his book Admiral of the Sea, 1942. Please see The Poincare Conjecture by Donal O’shea, 2007.

More Related