Introduction to Trigonometry: Grade 9

Transcription

1 Introduction to Trigonometry: Grade 9 Andy Soper October 6, 2013 This document was constructed and type-set using P C T E X (a dielect of L A T E X) 1

2 1 Before you start 1.1 About these notes. These notes are intended to be a gentle introduction to Trigonometry. I explore the the way the need for Trig. arose and how it was probably developed. I concentrate on the Greeks. Other civilizations in China and the Middle East also developed similar methods. The section on Ratios takes European Paper Sizes (A4, A3,... etc) as an example. If you find it complicated just skip that section. I have tried to cover the material slowly and with plenty of illustrations. Towards the end I have a short story about a Greek astronomer who visits Egypt for a holiday and decides to measure the height of the Great Cheops Pyramid. If you have any problem understanding nthis material or if you discover errors please click on (or paste to your web-browser s address bar) or to send me a message 2

3 1.2 What you will learn The Ancient Greeks wanted to find the length of a chord of a circle They had an angle and one side of a right triangle. Pythagoras could not help them here. They thought of similar s. Similar s have equal angles but different sides. However, the sides of each similar are always in the same ratio irrespective of the size of the similar s. s are similar if they have two corresponding angles equal. In right angled s only ONE angle is needed for similarity (the another is always 90 ) One angle, one side and the appropriate side ratios in a right-angled are sufficient to find the other two sides. These side-ratios are called Trigonometric (Triangle Measurement or Trig.) ratios We can find the Trig. ratios for a given angle in a right-angled by making a accurate sketch and measuring the sides. It does not need to be full-scale. Before we had pocket calculators people (even in antiquity) had tables of Trig. Ratios for angles to several places of decimals. Calculators use series (similar to number patterns) to calculate Trig. Ratios. 3

4 2 The Greeks seek an alternative to Pythagoras when only one side is known in a right-angled triangle. 2.1 An Old Problem The Ancient Greeks knew how to find a side of a right angles triangle if they already had two sides: Pythagoras Theorem. Figure 1: Pythagoras Theorem They also knew a lot about circles. They could find the perimeter (circumference) and the area, but they wanted to calculate the length of a chord. In the diagram the length of the chord is 2x and the angle subtended by half of the chord is 30. This angular value is convenient, as we shall see presently. Figure 2: The Chord Problem 4

5 They could make a right-angled Triangle by bisecting the chord (easily done with a compass). Another clever Greek, Euclid, told them that the line from the midpoint to the centre of the circle would make a right-angle at the chord. They knew one side, the radius, r and the angle 30, but Pythagoras could not help them here. Figure 3: Pythagoras Cannot Help Here. They turned to Similar Triangles. Figure 4: Similar Triangles have their Sides In Constant Ratio 5

6 2.2 Ratios Images, sheets of paper, TV and computer screens have what we call a Aspect Ratio. It is the ratio of width to height (short side to long side in the case of A4 paper). A piece of A4 paper is 210 mm by 297 mm and has an aspect ratio of 210 : 297 or = : 1 or 1 2 If you fold a piece of A4 paper in half along its long side (height) you get = or 1 2 (again) and we have A5 paper. If I double the width of a piece of A4 paper I get and we have A3 Paper. = guess?? good old In general if L is the long side and S is the short side L S = 2 L = S 2 Ratios allow us to make useful equations. If I double the short side it becomes the new long side L 2 = 2S and S 2 L 2 = L 2S = 2 2 and short long is again 2. The A series paper (ISO 216) retains its aspect ratio as it up-sizes and down-sizes. This means that if you make a A3 or A2 poster and you want to make an A5 hand-bill or flier the document will look the same, only smaller. We have similar rectangles! For your interest A2 is 420:594, A1 is 594:840 and A0 is 841:1189 (A0 has an area of 1 square metre. 6

7 2.3 The Greeks turn to Similar Triangles to find their chord They saw that similar triangles had their sides in constant ratio. This would apply to similar right-angled triangles also. Triangles are similar if two corresponding angles are equal. In the case of a right-angled triangle, the right-angle, 90, and one other angle are sufficient to prove similarity. The Ratios opposite hypotenuse adjacent hypotenuse opposite adjacent Figure 5: Similar Right-angled Triangles remain constant as long as the triagles remain similar (corresponding angles remain equal) 7

8 Figure 6: Try some of these similar Right-angled triangles: Opposite / Hypotenuse = 3/6 = 2.31/4.65 = 1.74/3.49 = 0.5 8

9 Now to return to the Greeks problem, Figure 7: Finding the length of the chord We have seen that in a right-angled triangle with one of the other angles being 30, the ratio of the side opposite the 30 angle to the hypotenuse is 0.5. Thus x R = 0.5 x = 0.5 R 9

10 2.4 Using these right-angled triangle ratios Figure 8: Naming the Ratios 10

11 Hipparchus (of Nicaea 190 BCE-120 BCE)the famous Greek Astronomer, was on holiday in Egypt and visited the Pyramids. Figure 9: The Great Pyramid of Cheops - Wikipedia Commons) He wondered how high the Great Pyramid was. He measured the length of the corner ridge in cubits (I have converted to metres for you) and he used his ancient astrolabe to find the angle of elevation of the top of the pyramid (42 degrees). Hipparchus is reputed to have invented the astrolabe for measuring the (angular) height of stars above the horizon. Now he calculated Figure 10: Hipparchus Measurements The modern value is m. h 218 = Sine(42 ) = h = 218 Sin(42 ) = 145.8m 11

12 2.5 A final question I hope that you are wondering how Hipparchus knew the value of Sine(42 )? All he had to do was make an accurate sketch of a right-angled with an angle of 42, measure the opposite side and hypotenuse, and divide the former by the latter like this: Figure 11: Sine of Trigonometry Then and Now We know that the ancient mathematicians made tables of Trig. ratios. When I was at High School a Calculator was a person who did calculations: We had books of tables of Trig. Ratios (and their Logarithms. When the British designed and built the rigid airship R101 in 1929 the internal stainless steel frame was designed by many human calculators using tables and slide-rules (a type of analog calculator using logarithms) Nothing much had changed since Hipparchus so far as everyday Trig was concerned. These tables contained thousands of values of the ratios for angles to several places of decimals. The people who created the tables certainly did not draw thousands of triangles and measure them. They used an algebraic calculation similar to this Taylor Series 12

13 Figure 12: Taylor Series for Sine (x) - Wikipedia Commons The series uses x in radians which requires that I divide 42 by 57.3 and get radians. The series seems to converge rapidly. Below I have calculated it to 3 terms Sin(0.733) = (0.733) (0.733) (0.733) = Our modern calculators probably use some similar formulae to generate Trig Ratios. 13