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

2. 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. 3 To convert an angle measured in degrees to radians do the following: Find the fraction of 360 0 the angle represents and multiply it by 2 π. So an angle of 30 0 is 30/360 * 2 π or 0.52 radians.

4. 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. 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 7.5 0 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, = 7.5 0 R, where R is the earth’s radius.

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

7. 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. 8 The line from you to the earth’s center has length r + b and forms the hypotenuse of a right triangle with legs r and a. The Pythagorean theorem gives: (r + b) 2 = r 2 + a 2 So r 2 + b 2 + 2br = r 2 + a 2 cancel r 2 on both sides. Since b is 30ft/5280ft, b 2 is very small, so set it to 0.

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

10. 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.