Some trigonometry problems

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Some trigonometry problems

• From Fred Greenleafs QR text
• Samuel Marateck

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Problem 1

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

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• 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 p. So an angle of 300 is 30/360 2 p
• or 0.52 radians.

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• 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 (its 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.

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• At noon the sun shone perpendicular to a
• tangent plane on the earths surface in
• Syene. In Alexandria, the suns 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 earths
• center. So s, which equals 500 miles,
• 7.50 R, where R is the earths radius.

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• 500 23.1459 7.5/360 R
• So R 500360/(23.14159 7.5)
• 3820 miles.

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Problem 2

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

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• The line from you to the earths 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 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.

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• So r2 b2 2br r2 a2 becomes
• 2br a2 .
• So a v(2br).
• r 3958 miles and b 30/5280, so
• a v (2303958/5280) v (45)
• a 6.7 miles

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• 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 Irvings description of Columbus
• as malicious nonsense in his book
• Admiral of the Sea, 1942. Please see The
• Poincare Conjecture by Donal Oshea, 2007.