Lets have a Math Chat Trigonometric Ratios l

Let’s have a Math Chat Trigonometric Ratios l Basic Terms about Angles l Definition of Trigonometric Ratios of Any Angles l Signs of Trigonometric Ratios 1

Basic Terms about Angles Suppose A is a point on the positive x-axis. When OA rotates about the origin O to OB, an angle θ is formed. y B angle of rotation terminal side θ O A x initial side Note: θ starts rotating from the positive x-axis. 2

Basic Terms about Angles When OA rotates about O anti-clockwise to OP, a positive angle is formed. y P Positive angle O A x Anticlockwise 3

Basic Terms about Angles When OA rotates about O clockwise to OS, a negative angle is formed. y O Clockwise S A x Negative angle 4

Basic Terms about Angles Two different angles may have the same initial side and terminal side. y O y x O x 5

Basic Terms about Angles The x-axis and the y-axis divide a rectangular coordinate plane into four regions. Each region is called a quadrant. y quadrant II x O quadrant III quadrant IV 6

Basic Terms about Angles We use the terminal side of angle �� to determine in which quadrant angle �� lies. y ∵ Terminal side lies in quadrant I. ∴ 70 lies in quadrant I. O A x When 0 < θ < 90 , �� lies in quadrant I. When 90 < �� < 180 , �� lies in quadrant II. When 180 < �� < 270 , �� lies in quadrant III. When 270 < �� < 360 , �� lies in quadrant IV. 7

Basic Terms about Angles Example 1 In the following, sketch 30 on a rectangular coordinate plane and name the quadrant in which the angle lies. y O x 30 lies in quadrant I. 8

Basic Terms about Angles Example 2 In the following, sketch 300 on a rectangular coordinate plane and name the quadrant in which the angle lies. y O x 300 lies in quadrant IV. 9

Basic Terms about Angles Example 3 In the following, sketch – 250 on a rectangular coordinate plane and name the quadrant in which the angle lies. y O x – 250 lies in quadrant II. 10

Basic Terms about Angles Example 4 Write down the quadrant in which each of the following angles lies. angle 75° 286° 189° – 120° – 300° 390° 470° quadrant I IV III I I II 11

Definition of Trigonometric Ratios of Any Angles We have learnt to define the trigonometric ratios of an acute angle �� with reference to a right-angled triangle. sin �� = r y cos �� = x tan �� = where 12

Definition of Trigonometric Ratios of Any Angles However, if the angle θ concerned is not an acute angle (e. g. θ is an obtuse angle), it is impossible to construct a right-angled triangle with this obtuse angle as shown in the figure below. Then how can we define trigonometric ratios of an angle which is not acute? 13

Definition of Trigonometric Ratios of Any Angles Let �� be an angle that has its initial side on the positive x -axis on a rectangular coordinate plane, and P(x, y) be a point on the terminal side of θ. The trigonometric ratios of θ are defined as follows: sin �� = y P (x, y) y r O cos �� = x x tan �� = r must take a positive value. The values of x and y could be positive, negative or 0. 14

Definition of Trigonometric Ratios of Any Angles Example 5 y P lies on the terminal side of an angle θ. Find the values of sin θ, cos θ and tan θ. (Leave the answers in surd form if necessary. ) a = – 4, b = 3. r = OP (– 4, 3) O � � x sin �� cos �� tan �� =5 15

Signs of Trigonometric Ratios How to determine the signs of trigonometric ratios of angles in different quadrants? Quadrant I y Quadrant II y P(a, b) O The ratio P(a, b) x O +/– + The ratio x +/– + + + 16

Signs of Trigonometric Ratios Quadrant III y Quadrant IV y O x P(a, b) The ratio x O P(a, b) The ratio Only tan θ is positive. 17

Signs of Trigonometric Ratios The signs of trigonometric ratios can be summarized in the “CAST” diagram. y Sin All Is positive Are positive x Tan Cos Is positive 18

Signs of Trigonometric Ratios Example 6 If 　　　　　 , where 270 < �� < 360 , find the values of sin �� and cos ��. (Leave your answers in surd form. ) ∵ 270 < �� < 360 ∴ �� lies in quadrant IV. Let P(a, b) be a point on the terminal side of ��. Let a = 5 and b = – 7. y � � x O (5, – 7) 19

Signs of Trigonometric Ratios Example 7 If 　　　　　 and cos �� < 0, find the values of cos �� and tan ��. (Leave your answers in surd form if necessary. ) y ∵ cos �� < 0 and sin �� > 0. ∴ �� lies in quadrant II. Let P(a, b) be a point on the terminal side of ��. Let b = and r = 2. � � O x 20