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**10 Trigonometry (1) 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 Home 10.6 Graphical Solutions of Trigonometric Equations

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**10.1 Basic Terminology of Trigonometry**

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. Fig. 10.3 Fig. 10.3 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.

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**10.1 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 Fig. 10.7 The x-axis and y-axis do not belong to any of the four quadrants.

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**10.1 Basic Terminology of Trigonometry**

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 360º, 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 Fig. 10.8 Since 120º lies in quadrant II, –240º also lies in quadrant II.

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**10.2 Definition of Trigonometric Ratio of Any Angles**

A. Definition The trigonometric ratios of an acute angle are defined as the ratios between the sides of a right-angled triangle. 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 Fig

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**10.2 Definition of Trigonometric Ratio of Any Angles**

For angles greater than 90o, we can consider the terminal side OP of the angle θ and let the coordinates of P be (x, y): θ lies in quadrant II θ lies in quadrant III θ lies in quadrant IV Fig (a) Fig (b) Fig (c) Then the trigonometric ratios of θ are defined in terms of x, y and r, as follows: where

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**10.2 Definition of Trigonometric Ratio of Any Angles**

The following identities hold for all measures of θ. (a) (b) Proof of (a): By Pythagoras’ Theorem, Proof of (b):

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**10.2 Definition of Trigonometric Ratio of Any Angles**

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. Notes: 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. Fig

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**10.3 Finding Trigonometric Ratios Without Using a Calculator**

A. Trigonometric Ratios of Angles Formed by the Coordinate Axes Fig (a) Fig.10.27(c) Fig.10.27(b) Fig.10.27(d)

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**10.3 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 10.3. θ sin θ cos θ tan θ 0º 1 90º undefined 180º –1 270º 360º 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

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**10.3 Finding Trigonometric Ratios Without Using a Calculator**

B. Trigonometric Ratios by Considering the Reference Angles 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). lies in quadrant I lies in quadrant III lies in quadrant II lies in quadrant IV Fig (a) Fig (c) Fig (b) Fig (d)

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**10.3 Finding Trigonometric Ratios Without Using a Calculator**

C. Finding Trigonometric Ratios when other Trigonometric Ratios are Given Example 10.3T If , where , find and . Solution: Since and θ lies in quadrant II, we take Since θ lies in quadrant II, then P(–2, 3) is a point on the terminal side of θ as shown in the figure.

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**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 . By definition,

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**10.4 Trigonometric Equations**

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.

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**10.4 Trigonometric Equations**

Example 10.5T If , where , find θ. Solution: Since tan θ < 0, θ must lie in either quadrant II or quadrant IV. Step 1: Step 2: Let β be the reference angle of θ. Then Step 3: By locating θ and its reference angle diagrammatically as shown in the figures, or

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**10.4 Trigonometric Equations**

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.

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**10.4 Trigonometric Equations**

Example 10.7T Solve the equation for (Give the answers correct to 1 decimal place.) Solution: If cos = 0, then sin = 1 , which does not satisfy 2sin + cos = 0. As cos 0, we can divide the expression by cos . (correct to 1 decimal place)

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**10.5 Graphs of Trigonometric Functions**

A. The Graph of y = sin x x 0o 30o 60o 90o 120o 150o 180o 210o 240o 270o 300o 330o 360o y 0.5 0.9 1 –0.5 –0.9 –1 Table 10.4 The ordered pairs in Table 10.4 are plotted and the graph of y = sin x is drawn and shown in Fig Fig

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**10.5 Graphs of Trigonometric Functions**

Properties of Sine Function: Fig Maximum value of sin x = 1 , which corresponds to x = 90o . Minimum value of sin x = –1 , which corresponds to x = 270o . The curve of y = sin x for repeats itself in every 360o.

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**10.5 Graphs of Trigonometric Functions**

B. The Graph of y = cos x x 0o 30o 60o 90o 120o 150o 180o 210o 240o 270o 300o 330o 360o y 1 0.9 0.5 –0.5 –0.9 –1 Table 10.5 Similarly, cos x repeats itself in every 360o. cos x is a periodic function with a period of 360o and for all values of x. Fig

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**10.5 Graphs of Trigonometric Functions**

C. The Graph of y = tan x x 0o 30o 60o 75o 90o 105o 120o 150o 180o y 0.6 1.7 3.7 undefined –3.7 –1.7 –0.6 x 210o 240o 255o 270o 285o 300o 330o 360o y 0.6 1.7 3.7 undefined –3.7 –1.7 –0.6 Table 10.6 Fig

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**10.5 Graphs of Trigonometric Functions**

Properties of tangent function : Fig , the tangent function exhibits the following behaviour: 1. For From 0o to 90o , tan x increases from 0 to positive infinity. From 90o to 180o , tan x increases from negative infinity to 0. 2. tan x is a periodic function with a period of 180o.

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**10.6 Graphical Solutions of Trigonometric Equations**

To solve the equation from the graph, draw a straight line as shown below. 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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