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Contents 10.1 Basic Terminology of Trigonometry 10.2 Definition of Trigonometric Ratio of Any Angles 10.3 Finding Trigonometric Ratios Without Using a.

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Presentation on theme: "Contents 10.1 Basic Terminology of Trigonometry 10.2 Definition of Trigonometric Ratio of Any Angles 10.3 Finding Trigonometric Ratios Without Using a."— Presentation transcript:

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2 Contents 10.1 Basic Terminology of Trigonometry 10.2 Definition of Trigonometric Ratio of Any Angles 10.3 Finding Trigonometric Ratios Without Using a Calculator 10.4 Trigonometric Equations 10.5 Graphs of Trigonometric Functions 10.6 Graphical Solutions of Trigonometric Equations Home 10 Trigonometry (1)

3 10 Home Content P.2 Fig A. Angles of Rotation Fig 10.3 shows a circle on a rectangular coordinate plane, with centre at the origin O and of radius r. A is a point on the circle Basic Terminology of Trigonometry An angle θ is formed by rotating OA about the origin O to reach OP as shown in Fig The angle θ is called the angle of rotation. We call OA the initial side of the angle and OP the terminal side of the angle. Fig. 10.3

4 Trigonometry (1) 10 Home Content P Basic Terminology of Trigonometry B. Quadrants In a rectangular coordinate plane, the x -axis and y -axis divide the plane into four regions which are known as quadrants. They are labelled and are shown in Fig The x -axis and y -axis do not belong to any of the four quadrants. Fig. 10.7

5 Trigonometry (1) 10 Home Content P.4 An angle of rotation θ is said to lie in a quadrant if its terminal side lies in that quadrant. When the measure of θ is either negative or greater than 360 º, the position of the terminal side of θ determines the quadrant in which it lies. For example, the terminal side of –240º coincides with that of 120º as shown in Fig Basic Terminology of Trigonometry Since 120º lies in quadrant II, –240º also lies in quadrant II. Fig. 10.8

6 Trigonometry (1) 10 Home Content P.5 A. Definition 10.2 Definition of Trigonometric Ratio of Any Angles To define the trigonometric ratios for an arbitrary angle, we can first place triangle OPQ on a rectangular coordinate plane as shown in Fig The trigonometric ratios of the angle  then can be defined in terms of x, y and r, where. The trigonometric ratios of an acute angle  are defined as the ratios between the sides of a right-angled triangle. Fig

7 Trigonometry (1) 10 Home Content P.6 θ lies in quadrant IIθ lies in quadrant IIIθ lies in quadrant IV 10.2 Definition of Trigonometric Ratio of Any Angles Then the trigonometric ratios of θ are defined in terms of x, y and r, as follows:where For angles greater than 90 o, we can consider the terminal side OP of the angle θ and let the coordinates of P be (x, y) : Fig (c)Fig (b)Fig (a)

8 Trigonometry (1) 10 Home Content P.7 The following identities hold for all measures of θ. (a) (b) Proof of (a): Proof of (b): 10.2 Definition of Trigonometric Ratio of Any Angles By Pythagoras’ Theorem,

9 Trigonometry (1) 10 Home Content P.8 B. Signs of the Three Trigonometric Ratios The signs of the three trigonometric functions can be summarized in Fig This figure is also called a ‘ASTC’ diagram. In the ‘ASTC’ diagram, the letters A, S, T and C indicate the trigonometric ratios that are positive, where A: All positives; S: Sin positives; T: Tan positives; and C: Cos positives. Notes: Fig Definition of Trigonometric Ratio of Any Angles

10 Trigonometry (1) 10 Home Content P.9 A. Trigonometric Ratios of Angles Formed by the Coordinate Axes Fig.10.27(b) Fig (a)Fig.10.27(c) Fig.10.27(d) 10.3 Finding Trigonometric Ratios Without Using a Calculator

11 Trigonometry (1) 10 Home Content P Finding Trigonometric Ratios Without Using a Calculator The values of the trigonometric ratios of,,, and can be found and the results are listed in Table θsin θcos θtan θ 0º0º º 10undefined 180º 0 –1– º–1–1 0undefined 360º 010 As shown in Fig (d), the terminal sides OP of and lie in the same position. Hence the trigonometric ratios of and are the same. Table 10.3

12 Trigonometry (1) 10 Home Content P.11 The positive acute angle formed between the terminal side of the angle  and the x -axis is called reference angle  as shown in Fig (a) – (d). B. Trigonometric Ratios by Considering the Reference Angles  lies in quadrant I  lies in quadrant II  lies in quadrant III  lies in quadrant IV Fig (b) Fig (a)Fig (c) Fig (d) 10.3 Finding Trigonometric Ratios Without Using a Calculator

13 Trigonometry (1) 10 Home Content P.12 C. Finding Trigonometric Ratios when other Trigonometric Ratios are Given 10.3 Finding Trigonometric Ratios Without Using a Calculator If, where, find and. Solution: Since θ lies in quadrant II, then P(–2, 3) is a point on the terminal side of θ as shown in the figure. Example 10.3T Since and θ lies in quadrant II, we take

14 Trigonometry (1) 10 Home Content P.13 By definition, 10.3 Finding Trigonometric Ratios Without Using a Calculator Consider (–2, 3) to be a point on the terminal side of θ. Then use the definitions to find other trigonometric ratios of .

15 Trigonometry (1) 10 Home Content P.14 A. Inverses of the Trigonometric Ratios If a trigonometric ratio of an angle is given, then we can find the angle. This process is called finding the inverse of the trigonometric ratio Trigonometric Equations

16 Trigonometry (1) 10 Home Content P.15 If, where, find θ. Solution: Since tan θ < 0, θ must lie in either quadrant II or quadrant IV. Step 1: Example 10.5T 10.4 Trigonometric Equations Step 3:By locating θ and its reference angle diagrammatically as shown in the figures, or Step 2:Let β be the reference angle of θ. Then

17 Trigonometry (1) 10 Home Content P.16 B. Algebraic Solutions of Trigonometric Equations An equation involving trigonometric ratios of an unknown angle is called a trigonometric equation. For example, (2 sin x – 1)(sin x + 1) = 0 is a trigonometric equation. It is not an identity because it holds for some values of x only, such as x = 30º. x = 30º is called a solution of this trigonometric equation Trigonometric Equations

18 Trigonometry (1) 10 Home Content P.17 (Give the answers correct to 1 decimal place.) Solution: (correct to 1 decimal place) If cos  = 0, then sin  =  1, which does not satisfy 2sin  + cos  = 0. As cos   0, we can divide the expression by cos . Example 10.7T 10.4 Trigonometric Equations Solve the equation for

19 Trigonometry (1) 10 Home Content P.18 A. The Graph of y = sin x x 0o0o 30 o 60 o 90 o 120 o 150 o 180 o 210 o 240 o 270 o 300 o 330 o 360 o y –0.5–0.9–1–0.9–0.50 The ordered pairs in Table 10.4 are plotted and the graph of y = sin x is drawn and shown in Fig Graphs of Trigonometric Functions Fig Table 10.4

20 Trigonometry (1) 10 Home Content P.19 Properties of Sine Function: Maximum value of sin x = 1, which corresponds to x = 90 o. Minimum value of sin x = –1, which corresponds to x = 270 o. The curve of y = sin x forrepeats itself in every 360 o Graphs of Trigonometric Functions Fig

21 Trigonometry (1) 10 Home Content P.20 B. The Graph of y = cos x x 0o0o 30 o 60 o 90 o 120 o 150 o 180 o 210 o 240 o 270 o 300 o 330 o 360 o y –0.5–0.9–1–0.9– Graphs of Trigonometric Functions Fig Table 10.5 Similarly, cos x repeats itself in every 360 o. cos x is a periodic function with a period of 360 o and for all values of x.

22 Trigonometry (1) 10 Home Content P.21 C. The Graph of y = tan x x 0o0o 30 o 60 o 75 o 90 o 105 o 120 o 150 o 180 o y undefined–3.7–1.7–0.60 x 210 o 240 o 255 o 270 o 285 o 300 o 330 o 360 o y undefined–3.7–1.7– Graphs of Trigonometric Functions Table 10.6 Fig

23 Trigonometry (1) 10 Home Content P.22 Properties of tangent function : From 0 o to 90 o, tan x increases from 0 to positive infinity. From 90 o to 180 o, tan x increases from negative infinity to tan x is a periodic function with a period of 180 o., the tangent function exhibits the following behaviour:1. For 10.5 Graphs of Trigonometric Functions Fig

24 Trigonometry (1) 10 Home Content P Graphical Solutions of Trigonometric Equations The straight line cuts the graph of at P and Q. The approximate solution is 228 or 312. To solve the equation from the graph, draw a straight line as shown below. The straight line does not intersect with the graph of.


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