3 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 3 TRIGONOMETRY 3.1. Radian and Arc Length The number π plays a central role in the study of trigonometry. We all know that a circle of radius 1 has area π and circumference 2 π . It is also very useful in describing angles, as we shall now show. Let us split a circle of radius 1 along a diameter into two semicircles as shown in the picture below. The circumference of the circle is now split into two equal parts, each of length π and each suntending an angle 180 . If we use the convention that π = 180 , then the arc of the semicircle of radius 1 will be the same as the angle it subtends. If we further split the arc of the semicircle of radius 1 into two equal parts, then each of the two parts forms an arc of length π/ 2 and subtends an angle 90 = 2. In fact, any arc of a circle of radius 1 which subtends an angle θ must have length θ under our convention. We now formalize our discussion so far. Definition. An angle of 1 radian is deFned to be the angle subtended by an arc of length 1 on a circle of radius 1. This chapter was written at Macquarie University in 1999.
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r θ r x y 1 3–2 W W L Chen and X T Duong : Elementary Mathematics Remarks. (1) Very often, the term radian is omitted when we discuss angles. We simply refer to an angle 1 or an angle π , rather than an angle of 1 radian or an angle of π radian. (2) Simple calculation shows that 1 radian is equal to (180 ) =57 . 2957795 ... . Similarly, we can show that 1 is equal to ( π/ 180) radian = 0 . 01745329 radian. In fact, since π is irrational, the digits do not terminate or repeat. (3) We observe the following special values: π 6 =30 , π 4 =45 , π 3 =60 , π 2 =90 = 180 , 2 π = 360 . Consider now a circle of radius r and an angle θ given in radian, as shown in the picture below. Clearly the length s of the arc which subtends the angle θ satisFes s = , while the area A of the sector satisFes A = πr 2 × θ 2 π = 1 2 r 2 θ. Note that 2 is equal to the area inside the circle, while θ/ 2 π is the proportion of the area in question. 3.2. The Trigonometric Functions Consider the xy -plane, together with a circle of radius 1 and centred at the origin (0 , 0). Suppose that θ is an angle measured anticlockwise from the positive x -axis, and the point ( x, y ) on the circle is as shown in the picture below. We deFne cos θ = x and sin θ = y. ±urthermore, we deFne tan θ = sin θ cos θ = y x , cot θ = cos θ sin θ = x y , sec θ = 1 cos θ = 1 x and csc θ = 1 sin θ = 1 y .
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2 -1 1 3 π 4 Chapter 3 : Trigonometry 3–3 Remarks. (1) Note that tan θ and sec θ are defned only when cos θ 6 = 0, and that cot θ and csc θ are defned only when sin θ 6 =0.
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3 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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