Angles of Rotation

Angle of Rotation Consider a line segment OA on a rectangular coordinate plane. If OA is rotated about the origin O to OB through an angle θ, then θ is called the angle of rotation from OA to OB. y B angle of rotation terminal side initial side θ x O A vertex

When the initial side OA is rotated in an anti-clockwise direction about O, the angle formed is a positive angle.

When the initial side OA is rotated in an clockwise direction about O, the angle formed is a negative angle.

rotate OA in an anti-clockwise direction Case 1 Case 2 rotate OA in an anti-clockwise direction rotate OA in a clockwise direction form a positive angle form a negative angle y y e.g. e.g. B anti-clockwise direction x O A –60 72 x clockwise direction O A C AOB is a positive angle. AOC is a negative angle.

Then the angle formed is greater than 360 or smaller than –360. Suppose the initial side OA is rotated through more than one complete turn to OP. Then the angle formed is greater than 360 or smaller than –360. For example, A positive angle greater than 360 A negative angle smaller than –360

The Four Quadrants The x-axis and the y-axis divide a rectangular coordinate plane into four quadrants. x O y Quadrant II Quadrant I Quadrant III Quadrant IV

For any angle θ between 0 and 360, the position of its terminal side determines the quadrant in which the angle lies.

II I IV III The terminal side lies in quadrant II. The terminal side lies in quadrant III. The terminal side lies in quadrant IV. 230 lies in quadrant III. 310 lies in quadrant IV.

In which quadrants do the following angles θ lie? (b) 315 (c) 62 (d) 119 y x O P 218 y x O P 315 O y x P 62 y x O P 119 218 lies in quadrant III. 315 lies in quadrant IV. 62 lies in quadrant I. 119 lies in quadrant II.

Summary of quadrant in which θ lies 90 < θ < 180 0 < θ < 90 Quadrant II Quadrant I x Quadrant III O Quadrant IV 180 < θ < 270 270 < θ < 360

In which quadrants do negative angles and angles greater than 360 lie? For these angles, the position of the terminal side determines the quadrant in which the angle lies. For example, An angle of 510 lies in quadrant II. An angle of –70 lies in quadrant IV.

In which quadrants do the following angles lie? θ 390 470 –140 –80 Position of the terminal side of θ Quadrant in which θ lies x y O P 390 y x O P 470 –140 y x O P y x O –80 P Quadrant I Quadrant II Quadrant III Quadrant IV

In which quadrants do the angles 90, 180, 270 and 360 lie? They do NOT lie in any quadrants, but 90 and 270 lie on the y-axis, 180 and 360 lie on the x-axis.

The terminal sides of each pair of angles coincide. Do you notice anything special in the following pairs of angles? 30 and 390 150 and 510 220 and –140 290 and –70 y O P 30 x y x O Q 150 y x O R 220 y x O S 290 –140 –70 510 390 The terminal sides of each pair of angles coincide.

Let’s study each pair of angles. x OP is the terminal side of both 390 and 30 30 390 = 360 + 30

OQ is the terminal side of both 150 and 510 y x O Q OQ is the terminal side of both 150 and 510 150 510 = 360 + 150 y x O R OR is the terminal side of both 220 and –140 220 –140 = 220 – 360

OS is the terminal side of both 290 and –70 y x O S 290 OS is the terminal side of both 290 and –70 –70 = 290 – 360 However, two angles with the same terminal sides may NOT be equal angles. For example, 290 and –70 have the same terminal sides, but 290 ≠ –70. Angles with the same terminal sides differ by 360 or multiples of 360.

Follow-up question In which quadrant does the angle –30 lie? Determine whether –30 and 330 have the same terminal side. y 330 x O –30 = 330 – 360 (i) 30 lies in quadrant IV. (ii) –30 and 330 have the same terminal side.

Follow-up question In which quadrant does the angle 110 lie? Determine whether 110 and 470 have the same terminal side. y 110 470 x O = 360 +110 (i) 110 lies in quadrant II. (ii) 110 and 470 have the same terminal side.

Trigonometric Ratios of Any Angle

Definitions of Trigonometric Ratios of Any Angle Let’s review the trigonometric ratios for acute angle θ.

Consider a right-angled triangle POQ. opposite side = hypotenuse sin q r y = r y = hypotenuse adjacent side cos q x r = θ = adjacent side opposite side tan q x y = O Q x

We have learnt about trigonometric ratios for acute angle θ only. What if θ > 90 or θ < 0? We can extend the definition of trigonometric ratios of any angle as follows.

When △POQ is placed on a rectangular coordinate plane, (x, y) are the coordinates of P and r is the length of OP. r y = q sin length of OP O y x Q P x y r θ (x, y) r x = q cos y-coordinate of P x y = q tan x-coordinate of P By Pythagoras’ theorem, r = 2 y x + θ in quadrant I

, 90º < θ < 180º 180º < θ < 270º 270º < θ < 360º Consider 90 < θ < 360. Rotate the terminal side OP of θ about the origin to , quadrant IV quadrant III quadrant II 90º < θ < 180º 180º < θ < 270º 270º < θ < 360º O y x O y x O y x r P(x, y) θ θ θ r P(x, y) r P(x, y) The trigonometric ratios of θ are defined as follows: r y = q sin x cos tan , , where > 0 2 +

In fact, the trigonometric ratios of any angle θ are defined by the coordinates of P and the length of OP, i.e. (x, y) and r, as follows: r y = q sin x cos tan , , where > 0 2 +

90º < θ < 180º 180º < θ < 270º 270º < θ < 360º x < 0 y > 0 O y x O y x O y x r P(x, y) θ θ θ r P(x, y) r P(x, y) x < 0 y < 0 x > 0 y < 0 Since x and y can be positive, negative or zero, the trigonometric ratios of θ can also be positive, negative or zero.

Find the values of sin θ, cos θ and tan θ in the figure. y x θ P(–1, 2) O x = -1 x = –1 ∵ , y = 2 5 2 ) 1 ( = + - r OP ∴ r sin = r y q ∴ y = 2 ) 5 2 (or =

Find the values of sin θ, cos θ and tan θ in the figure. y x θ P(–1, 2) O x = –1 ∵ , y = 2 5 2 ) 1 ( = + - r OP ∴ cos = r x q ∴ ) 5 (or 1 - =

Find the values of sin θ, cos θ and tan θ in the figure. y x θ P(–1, 2) O x = –1 ∵ , y = 2 5 2 ) 1 ( = + - r OP ∴ tan = x y q ∴ 1 2 - = 2 - =

Follow-up question In the figure, P (–8, –6) is a point on the terminal side of q . Find the values of sin q, cos q and tan q. y ∵ x = –8, y = –6 q ∴ OP x O P(–8,–6)

Trigonometric Ratios of 0º, 90º, 180º, 270º and 360º How about the trigonometric ratios of some special angles, including 0º, 90º, 180º, 270º and 360º? I have learnt how to find the trigonometric ratios of angles in different quadrants.

Since the terminal sides of 0º and 360º coincide, we have (a) When θ = 0º, x = r , y = 0  P(r, 0) θ = 0º O y x sin = º r y  x = r y = 0 1 cos = º r x tan = º r x y Since the terminal sides of 0º and 360º coincide, we have sin 360º = sin 0º = 0 cos 360º = cos 0º = 1 tan 360º = tan 0º = 0

∵ The denominator is zero. (b) When θ = 90º,  x = 0 , y = r P(0, r) θ = 90º O y x x = 0 y = r 1 r 90 sin = º y  90 cos = º r x r 90 tan (undefined) = º x y ∵ The denominator is zero.

 (c) When θ = 180º, x = –r , y = 0  P(–r, 0) θ = 180º O y x 180 sin 180 sin = º r y  x = –r y = 0 –1 180 cos = º r –r x 180 tan = º –r x y

∵ The denominator is zero. (d) When θ = 270º, P(0, –r) θ = 270º O y x x = 0 , y = –r  –r –1 sin 270º = r y  cos 270º = r x x = 0 y = –r –r tan 270º (undefined) = x y ∵ The denominator is zero.

Trigonometric ratios of 0º, 90º, 180º, 270º and 360º θ 0º 90º 180º 270º 360º sin θ cos θ tan θ Trigonometric ratio 1 1 undefined –1 –1 undefined 1

Evaluate the following expressions without using a calculator. Follow-up question Evaluate the following expressions without using a calculator. (a) (b) (c) (d) As is undefined, is undefined.

The Signs of Trigonometric Ratios The sign of a trigonometric ratio depends on which quadrant the angle lies in. Consider the 4 cases below. q in quadrant I q in quadrant III q in quadrant II q in quadrant IV

> 0 < 0 > 0 < 0 > 0 > 0 < 0 Angle  lies in Quadrant I Quadrant II Quadrant III Quadrant IV x y r sin  cos  tan  O x y P(x, y)  O x y P(x, y)  O x y P(x, y)  O x y P(x, y)  × × × × > 0 < 0 > 0 < 0 > 0 > 0 < 0 > 0 > 0 < 0 < 0 > 0 < 0 < 0 > 0 > 0 < 0 > 0 < 0

Quadrant in which θ lies Quadrant II Quadrant III Quadrant IV sin θ > 0 < 0 cos θ tan θ In which quadrants are sin θ, cos θ and tan θ positive? sin θ: quadrants I and II cos θ: quadrants I and IV tan θ: quadrants I and III

Summary: ‘CAST’ diagram Only Sine ratio is positive I IV II III O Cos All Sin Tan y x All trigonometric ratios are positive Only Tangent ratio is positive Only Cosine ratio is positive

Follow-up question Determine in which quadrant each of the following angles lies and then indicate the signs of its trigonometric ratios. (a) 110 (b) 350 y y P S A S A 350 110 x x O O P T C T C ∵ 110 lies in quadrant II. sin 110 is positive. ∴ ∵ 350 lies in quadrant IV. sin 350 is negative. ∴ cos 110 is negative. cos 350 is positive. tan 110 is negative. tan 350 is negative.

If one of the trigonometric ratios of an angle is given, we can use the definitions of trigonometric ratios and the ‘CAST’ diagram to find the other two trigonometric ratios.

Let us see the following example.

Given that sin q = and 270 < q < 360, find the values of cos q and tan q . 5 3 - ∵ 270 < q < 360 O y x ∴ q lies in quadrant III. θ

Let P(x, y) be a point on the terminal side of q and OP = r. Given that sin q = and 270 < q < 360, find the values of cos q and tan q . 5 3 - Let P(x, y) be a point on the terminal side of q and OP = r. r y 5 3 = - ∵ sin q = O y x Let y = -3 and r = 5. θ r 5 P(-4, -3) P(x, -3) P(x, y) x < 0 y < 0 x = -4 or x = 4 (rejected) r x x y ∴ cos q = and tan q =