Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Chapter 8 Therefore, the sine, cosine and tangent can also be defined as: Opposite Adjacent Opposite sin θ = , cosθ = , tan θ = Hypotenuse Hypotenuse Adjacent Trigonometry It can be memorized by the mnemonic “oh, ah, oh-ah”, which represents, sin θ is opposite over hypotenuse, cos θ is adjacent over hypotenuse, tan θ is opposite over adjacent. Trigonometry (三角學 三角學) 三角學 Definition of Angle Trigonometry is a branch of geometry that studies the relationship of triangles. Take a simplest triangle – a right-angled triangle as example: In plane geometry, the range of angle is 0° to 360°. However, this is not enough for practical use. For example, a clock rotates more than 360°. It is necessary for us to redefine what is angle. A b c θ B a C If we fix point B and C (the value of a), and vary the size of ∠BCA (value of θ), the value of b and c also changes. Actually, the value c/b is a function of θ. Trigonometry defined the function “sine” (正弦) as followed: c sin θ = b Similarly, there are also five other functions – cosine (餘弦), tangent (正切), cotangent (餘 切), secant (正割) and cosecant (餘割), together called the trigonometric functions (三角 函數): a c a b b cosθ = , tan θ = , cot θ = , secθ = , cscθ = b a c a c For practical use, only sine, cosine and tangent are more useful. In trigonometry, we interpret angle as rotation of rays (射線). When a ray OA rotates about O to the position OB, the amount of rotation is defined as the angle AOB and is denoted by ∠AOB . The angle is positive if it is rotated anti-clockwise, and is negative if is rotated clockwise. B 420° -75° A Hypotenuse 45° Opposite θ O B Adjacent C Respecting to θ, the side opposite to it is called the “opposite” (對邊), the one which is longest is called “hypotenuse” (斜邊), and the one next to θ is called “adjacent” (鄰邊). A Base on the “new” definition of angle, we also have a more generalized definition of trigonometric functions. 46 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry The radian measure of an angle is the ratio of the arc, which subtends the angle at the center of a circle, and the radius of the circle. Or in formula, s θ= r The following angle has a size of 1 radian. In a coordinate plane, take a point P (x, y). Let r denotes the distance from this point (P) to the origin O (0, 0). (So r = x + y .) Let θ be the angle from the x-axis to OP. 2 2 P (x, y) r r θ 1 rad. O Then we define: r Usually the unit “rad.” is omitted, i.e., the size of the angle is 1. y x cosθ = r r y x tan θ = cot θ = x y r r secθ = cscθ = x y These are the definition of the trigonometric functions of general angles θ. sin θ = Note that: π radians = 180° s r In elementary mathematics, we measure angles in degrees (角度). Say, a right angle is 90°. However, it is sometimes more convenient to use another measure: radians. A r Area = s θ r θ r In a sector (扇形), if θ is measured in radian, then: s = rθ Radians (弧度 弧度) 弧度 O r B In the figure, O is the center of a circle with radius r. A and B are two points on the circumference. θ is the size of ∠AOB . s is the length of arc AB. 47 1 r 2θ 2 = 12 rs Advanced Mathematics Training Class Notes Chapter 8: Trigonometry 30° (π/6) 1 2 Basic Trigonometric Relations sin Base on the definition of trigonometric functions, it is always true that: 1 cscθ = sin θ 1 1. secθ = (Reciprocal relations 倒數關係) cos θ 1 cot θ = tan θ sin θ tan θ = cosθ (Quotient relations 商數關係) 2. cos θ cot θ = sin θ 2 sin θ + cos 2 θ = 1 3. 1 + tan 2 θ = sec 2 θ cos tan 3 2 or 3 3 0.75 45° or 0.5 or 1 2 or 0.5 or 0.75 1 2 1 3 sin 0 2 1 2 2 2 3 2 4 2 cos 4 2 3 2 2 2 1 2 0 2 The followings should also be memorized: 0° (0) 90° (π/2) 180° (π) 270° (2π/3) 360° (2π) 1 0 -1 0 sin 0 0 -1 0 1 cos 1 / 0 / 0 tan 0 “/” means undefined. Some special angles 1 or 3 2 0° (0) 30° (π/6) 45° (π/4) 60° (π/3) 90° (π/2) The last relations can be proved by Pythagoras’ theorem. 2 3 60° (π/3) 1 2 Together with 0° and 90°, the value of sine and cosine shows an interesting pattern. (Squares relations 平方關係) 2 2 2 1 3 or 1 + cot 2 θ = csc 2 θ 1 45° (π/4) 2 2 Extension: sin18° = 60° 1 For 30°, 45° and 60°, their corresponding trigonometric functions will give some special results. They are thus called the special angles. The results should be memorized, and is summarized in the following table: 48 1 1+ 5 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Ranges of Trigonometric Functions Transformation Formulae The output ranges of some trigonometric functions are constrained: −1 ≤ sin θ ≤ 1 −1 ≤ cos θ ≤ 1 secθ ≤ −1 or secθ ≥ 1 Any angles can actually transformed into angles within range of 0° to 90° when dealing with trigonometric functions. Negative angles cscθ ≤ −1 or cscθ ≥ 1 But tangent and cotangent are not. Their output range is any real numbers. When θ = nπ + π 2 sin ( −θ ) = − sin θ cos ( −θ ) = cosθ tan ( −θ ) = − tan θ , tangent and secant do not exist. When θ = nπ, cotangent and cosecant are not defined. Complementary angles In addition, their signs are also arranged in a special pattern: 0° < θ < 90° 90° < θ < 180° 180° < θ < 270° 270° < θ < 360° 0 < θ < π /2 π/2 < θ < π π < θ < 3π/2 3π/2 < θ < 2π + + – – sin, csc + – – + cos, sec + – + – tan, cot ( π2 − θ ) = cosθ cos ( π2 − θ ) = sin θ tan ( π2 − θ ) = cot θ = tan1 θ sin (Recall: π/2 = 90°) It can be simplified as: 90° S Multiple revolutions f ( 2π n + θ ) = f (θ ) A 180° Where f can be sin, cos, tan, cot, sec or csc. n can be any integers. Moreover, tan(πn + θ) = tan θ, cot(πn + θ) = cot θ. 0° / 360° T C Others 270° 90° - θ 90° + θ 180° - θ 180° + θ 270° - θ 270° + θ 360° - θ π/2 - θ π/2 + θ π-θ π+θ 3π/2 - θ 3π/2 + θ 2π - θ -s -c -c -s s c c s sin -s -c -c -s s c c s cos -k -t t k -k t t k tan Here, “s” means “sin θ”, “c” means “cos θ”, “t” means “tan θ” and “k” means “cot θ”. θ “S” means “sine in positive”, “C” means “cosine is positive”, “T” means “tangent is positive”, “A” means “all are positive”. This is known as the “CAST Diagram”. Note: the part 0° to 90° is called quadrant I (第一象限), 90° to 180° is called quadrant II (第二象限), and so on. 49 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Double angle formulae (倍角公式 倍角公式) 倍角公式 Compound Angle Formulae (複角公式 複角公式) 複角公式 Replacing B with A, and use “+” for the “+” signs, we get the double angle formulae: sin 2 A = 2sin A cos A We often encounter expressions like sin(A + B) or cos(A – B). If A is a multiple of π/2, then we can use the transformation formulae. But it’s not that easy. Fortunately, there are general expressions of the trigonometric functions of sum or differences of angles. sin ( A ± B ) = sin A cos B ± cos A sin B cos 2 A = cos 2 A − sin 2 A tan 2 A = cos ( A ± B ) = cos A cos B ∓ sin A sin B cot 2 A − 1 2cot A For cosine, if we replace either cos2 A by 1 – sin2 A or sin2 A by 1 – cos2 A, we get two other forms of double angle formulae for cosine: cos 2A = 1 – 2 sin2 A = 2 cos2 A – 1 cot 2 A = tan A ± tan B tan ( A ± B ) = 1 ∓ tan A tan B cot A cot B ∓ 1 cot ( A ± B ) = cot A ± cot B These are called the “Compound Angle Formulae”. The first three formulae should be memorized. Triple angle formulae (三倍角公式 三倍角公式) 三倍角公式 Example: Express tan 15° in surd form (根式) (with root signs) Notice that tan 15° = tan (60° - 45°). Therefore, tan15° = tan ( 60° − 45° ) Again, using compound angle formulae, we get the triple angle formulae: sin 3 A = 3sin A − 4sin 3 A cos3 A = 4cos3 A − 3cos A The triple angle formulae can also be: tan 60° − tan 45° 1 + tan 60° tan 45° 3 −1 = 3 +1 = = 2 tan A 1 − tan 2 A sin 3 A = 3cos 2 A sin A − sin 3 A cos3 A = cos3 A − 3cos A sin 2 A But these are not as important as the two formulae given above. 3 −1 3 −1 3 + 1 3 −1 Half angle formulae (半角公式 半角公式) 半角公式 =2− 3 Finally, the half angle formulae states that: 1 − cos A A sin = ± 2 2 1 + cos A A cos = ± 2 2 The sign is determined by which quadrant does A/2 lie. 50 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Subsidiary angle form Product and Sum of Sine and Cosine Product-to-sum and sum-to-product mainly deal with sum or difference of two sines or cosines. The subsidiary angle form is much difference. It changes expression in form of a sin θ + b cosθ to r cos(θ – α). Suppose r cos (θ − α ) = a sin θ + b cosθ . By expanding the left-hand-side, comparing When solving trigonometric equations, we often come across with products or sums of sine and cosine, which we may not be able to do easily. The product-to-sum formulae, sum-to-product formulae and subsidiary angle form help us to tackle them. coefficients, we get: Product-to-sum formulae (積化和差公式 積化和差公式) 積化和差公式 a = r sin α b = r cos α ( sin ( A + B ) + sin ( A − B ) ) cos A sin B = ( sin ( A + B ) − sin ( A − B ) ) cos A cos B = ( cos ( A + B ) + cos ( A − B ) ) sin A sin B = − 12 ( cos ( A + B ) − cos ( A − B ) ) sin A cos B = 1 2 1 2 1 2 And therefore: r = a2 + b2 tan α = Example: If sin θ + 3 cosθ = r cos (θ − α ) , find r and α. Sum-to-product formulae (和差化積公式 和差化積公式) 和差化積公式 r = 12 + ( ) cos ( ) sin A − sin B = 2cos ( A+2 B ) sin ( A−2 B ) cos A + cos B = 2cos ( A+2 B ) cos ( A−2 B ) cos A − cos B = −2sin ( A+2 B ) sin ( A−2 B ) sin A + sin B = 2sin A+ B 2 A− B 2 tan α = A+ B 2 s c c s c s c s A− B 2 = = = = s s c c + − + − s s c c A B Here, “s” means “sin”, “c” means “cos”. 51 ( 3) 2 =2 1 3 ∴ α = π6 The product-to-sum and sum-to-product formulae can be summarized as: A B A+B A–B 2 2 2 −2 a b ( = 30° ) Advanced Mathematics Training Class Notes Chapter 8: Trigonometry y Graphs of Trigonometric Functions (三角函數的圖像 三角函數的圖像) 三角函數的圖像 2 y = cot x y 1 x y = sin x −π 0 π 2π x −π 0 π -2 2π y -1 2 y = sec x y 1 x y = cos x −π 0 π 2π x −π 0 π -2 2π y -1 2 y = csc x y x 2 −π y = tan x 0 x −π 0 π -2 2π -2 52 π 2π Advanced Mathematics Training Class Notes Chapter 8: Trigonometry The previous page shows the graphs of the six trigonometric functions. In their graphs, we see that they have repeating patterns. They are then called “periodic functions” (週期函數). y 1 A function f(x) is periodic if and only if, there exists a real constant T > 0 such that: f(x + T) = f(x) for all real values of x. The smallest possible value of T is called the “period” (週期) of that function. x 0 π 2π 3π 4π For sine, cosine, secant and cosecant, their periods are 2π. For tangent and cotangent, their periods are π. Sine curve (正弦曲線 正弦曲線) 正弦曲線 -1 (Blue curve: y = sin x. Green curve: y = sin 2x . Purple curve: y = sin 2x) The graph of y = sin(x) is exactly the same as the shape of waves. By this, we will go depth in this type of curve. y 1 x−d The sine curve can be generalized as y = a sin + c . In this expression, a is called the T “amplitude” (振幅) of the wave, T is the period, c presents the horizontal translation (橫 移), and d illustrates the vertical translation (縱移). x 0 π 2π Note: The reciprocal (倒數) of T, 1/T, is sometimes called “frequency” (頻率) -1 The following graphs show how these four variables control the outlook of the curve. ( ) ( ) (Blue curve: y = sin x. Green curve: y = sin x − π4 . Yellow curve: y = sin x + π5 .) y y 2 2 x x −π 0 π 2π 0 π 2π -2 -2 (Green curve: y = sin x. Red curve: y = 2 sin x. Pink curve: y = − 12 sin x ) (Blue curve: y = sin x. Green curve: y = sin x – 1. Purple curve: y = sin x + 32 .) 53 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Properties of inverse trigonometric function Inverse Trigonometric Functions (反三角函數 反三角函數) 反三角函數 By the principle of inverse function, there must be: ( ) cos ( cos x ) = x tan ( tan x ) = x Inverse function (反函數 反函數) 反函數 For any function f(x), if y = f(x), there is a corresponding “inverse function” F(x) such that x = F(y). y+6 For example, if f(x) = 5x – 6, then its inverse function is F ( y ) = . 5 -1 An inverse function of f(x) is usually denoted as f (x). In the above example, x+6 . f −1 ( x ) = 5 sin sin −1 x = x sin −1 ( sin x ) = x −1 cos −1 ( cos x ) = x −1 tan −1 ( tan x ) = x Deriving from the relation #3 (Squares relations), we get: ( ) ( ) sec ( tan −1 x ) = csc ( cot −1 x ) = tan ( sec−1 x ) = cot ( csc −1 x ) = sin cos −1 x = cos sin −1 x = 1 − x 2 Inverse trigonometric function Being functions, trigonometric functions also have their inverse functions. They are denoted as sin-1 x, cos-1 x, etc. However, since trigonometric functions are all periodic, if y = f(x) (f is a trigonometric function), there are multiple solutions of x. For example, if 12 = sin x , then Or summarizing: x sin-1 x = − 53π , − 56π , − 23π , π6 , π3 , 76π , 43π , 136π ,… . But recall that a function can have one and only one corresponding value of dependant variable. So we must take only one value out of so many values. Which should we take? First, we define the principle value intervals (主值區間) of these inverse functions. − π2 ≤ sin −1 x ≤ π2 cos-1 tan-1 0 ≤ cos −1 x ≤ π − π2 < tan −1 x < π2 cot-1 As π/6 is the only value inside range –π/2 to π/2, so sin −1 12 = π6 . sec-1 sin cos 1 − x2 x 1 − x2 x x 1 1+ x 2 1 1+ x 1+ x x2 − 1 x 1+ x 1 x 2 cot 1− x 2 1− x x 2 1− x x x 2 x2 − 1 sec 1 1− x 1 − x2 1 x csc 2 1 x 1 1 − x2 2 x 1 x 1 + x2 1 + x2 x 2 1 x x 1 + x2 x 1 + x2 x 2 tan x 1 + x2 x2 − 1 1 1 2 x −1 x x x 2 x −1 x −1 x x2 − 1 x2 − 1 x2 − 1 x (The order of reading is first function, then inverse function, then “x”, and finally the cell) csc-1 54 1 x Advanced Mathematics Training Class Notes Chapter 8: Trigonometry General Solutions of Trigonometric Functions (三角函數通解 三角函數通解) 三角函數通解 Slope and Trigonometry = sin x , there are infinitely many solutions for x. Sometimes we are Recall the definition of slope of a line: We have learnt that if 1 2 m= told to find all of the solutions of the equation. Of course, we can’t list out that infinitely many solutions. We can use an expression with a free variable to represent it. The expression is called the general solution of a trigonometric function. C (x1, y1) If s = sin x, then −1 x = nπ + ( −1) sin s ( In radian measure ) x = 180°n + ( −1) sin −1 s n ( In degree measure ) x = 2nπ ± cos −1 c ( In radian measure ) ( In degree measure ) n If c = cos x, then −1 x = 360°n ± cos c A (x2, 0) B (x1, 0) If the second point lies on the x-axis (y2 = 0), the equation is reduced to m = x = nπ + tan −1 t ( In radian measure ) ( In degree measure ) −1 x = 180°n + tan t = sin x , then x = nπ + ( −1) nπ 6 y1 . x1 − x2 In the figure, AB = x1 − x2 BC = y1 In the above expressions, n is any integer. 1 2 θ x If t = tan x, then Example: y1 − y2 x1 − x2 Therefore, in ∆ABC, BC y1 tan θ = = =m AB x1 − x2 That means, , where n is any integer. In addition, if a range 0 < x < 2π (i.e., 0° < x < 360°) is specified, then x = sin −1 s or π − sin −1 s x = cos −1 c or 2π − cos −1 c x = tan −1 t or π + tan −1 t m = tan θ Where θ is the angle between the line and the x-axis, called the inclination (傾角) (180° − sin 1 s ) ( 360° − cos 1 c ) (180° + tan 1 t ) − − Example: If slope of a line is 1, its inclination is 45°. − 55 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Angle between two lines Revision Consider two lines L1 and L2 having slopes m1 and m2 and inclination θ1 and θ2. We are interested in the angle between them. In this chapter, we’ve learnt: 1. What are trigonometric functions 2. Definition of general angles 3. Radian measure of angles 4. Basic trigonometric relations 5. Trigonometric functions of some special angles 6. Ranges of trigonometric functions 7. Transformation formulae 8. Compound angle formulae 9. Product-to-sum and sum-to-product formulae, subsidiary angle form 10. Graphs of trigonometric functions 11. Inverse trigonometric functions 12. General solutions of trigonometric functions 13. Relationships between slope and tangent 14. Angle between two lines on coordinates plane φ L1 L2 θ1 ∵ φ = θ1 − θ 2 θ2 ( ext. ∠ of △ ) ∴ tan φ = tan (θ1 − θ 2 ) = tan θ1 − tan θ 2 1 + tan θ1 tan θ 2 = m1 − m2 1 + m1m2 Where φ is the angle between L1 and L2. Thus we got the formula: tan φ = m1 − m2 1 + m1m2 The absolute sign is there to ensure that an acute angle is selected. Example: Find the angle between two lines with slope = -2 and 3 respectively. tan φ = −2 − 3 1 + ( −2 )( 3) =1 ∴ φ = 45° 56 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry 89 tan k ° ∑ 1 + tan k ° Exercise 9. (PCMSIMC 2001) Evaluate In the followings, if not specified, x is the variable. Give answer in radians unless otherwise stated. If general solution is required, use “n” for the free integral variable. 10. (HKCEE 1995 Partial) Answer the following questions. a) Show that cos2 A – cos2 B = sin(A + B) sin(A – B) b) Show that cos2 x – sin2 y = cos(x + y) cos(x – y) k =1 c) Find the general solutions of cos 2 2 x + cos x sin 5 x − sin 2 3 x = 0 11. Prove that (sin x + cos x)2 = 1 + sin 2x [Note: This identity worth memorizing] n −1 2kπ 12. Prove that ∑ cos x + =0. n k =0 1. Express sin 75° the followings in surd form. 2. Find the general solutions of x in the following equations: a) sin x + cos x = 1 b) 3 – cos2 x – 3 sin x = 0 ( ) c) sin 4 x − π3 = cos ( π −312 x ) n −1 3. Identify the amplitude, period, horizontal and vertical shift of the following sine curve: 13. Prove that ∑ sin x + k =0 2 x −2π −π 0 π 2π -2 4. Given θ = 54°. a) Prove that cos 2θ + sin 3θ = 0. b) Prove the sin 3β = 3 sin β – 4 sin3 β. c) Show that sin 54° is a root of the equation 4x3 + 2x2 – 3x – 1 = 0. d) Find the exact value of sin 54°. [Hint: Consider the sign of sin 54°] 5. (HKMO 2000 Heat) How many roots of θ are there in (cos2 θ – 1) (2 cos2 θ – 1) = 0, where 0° < θ < 360°? 6. (HKMO 1998 Heat) The circumference of a circle is 14π cm. Let X cm be the length of an arc of the circle, which subtends an angle of 1/7 radian at the center. Find X. 7. (HKMO 2002 Heat) Given that ∠A is a right angle in ∆ABC, sin 2 C − cos 2 C = 1 4 and AB = 40 . Find the length of BC. 8. (PCMSIMC 2001) If sin x° = cos x°, where 0 < x < 360, find the sum of all possible values of x. 57 2kπ n =0 14. If tan a and tan b are roots of x2 + 2x + 3 = 0, find tan (a + b). 15. The vertices of ∆ABC are A (2, -1), B (7, 1) and C (-1, 5). Find tan A + tan B + tan C. 16. (IMO Prelim HK 2003) Find (cos 42° + cos 102° + cos 114° + cos 174°)2. y 4 Advanced Mathematics Training Class Notes Chapter 8: Trigonometry Suggested Solutions for the Exercise 1) 2+ 6 4 2a) 2π n + π2 b) nπ + ( −1) c) or 2π n nπ 2 1 7π π n + 4 12 3) a = 3, T = 1/2, c = π/2, d = 1. 5 +1 4d) 4 5) 5 6) 1 7) 8 8) 270 9) 44.5 π 10c) 2nπ ± π2 or n5π − 20 14) 1 15) 111/8 16) 3/4 58