00 (3) 2016 - 2017 H2 Maths Trigonometry Notes (student)

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 1 of 11 National Junior College 2016 – 2017 H2 Mathematics Revision: Trigonometry Key Questions to Answer:  What are the three basic and three non-basic trigonometric functions?  What is meant by the basic or reference angle?  How do I determine the sign (+ or –) of a trigonometric expression expressed either in degree or radians?  What are the key characteristics of the graph of the three basic trigonometric functions?  Why is it important to define principal values or principal range of inverse trigonometric functions?  Are inverse trigonometric functions the same as the reciprocal of trigonometric functions?  What are the trigonometric values of special angles  30 ,45 ,60   ?  How do you obtain the value of a trigonometric function from another, i.e., g iven sin 0.4A where A is an acute (or obtuse) angle, what is cos ?A  What are some uses for each of the following:  Pythagorean identities,  Addition Formulae,  Double Angle Formulae,  Sum-to-Product/Factor Formulae,  Product-to-Sum Formulae,  R-formulae?  In what situations do we use the sine/cosine rule to fi nd the angle and/or length of one side of a triangle? §1 Introduction Waves occur in many natural phenomena such as seismic waves in earthquakes and ocean waves. Due to their periodic behavior, these waves are commonly modeled by the two trigonome tric functions, sine and cosine. In one computer generated visual simulation, it showed that MV Derbyshire, the biggest British registered merchant ship ever to have been lost at sea, was sunk by typhoon Orchid because the ocean waves at the time of incident were just slightly longer than 294 metres, the length of the ship. This meant that her bow would have been submerged before she had the chance to rise above the waves. Source: http://www.liverpoolmuseums.org.uk/maritime/exhibitions/derbyshire/

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 2 of 11 §2 The Three Basic Trigonometric Functions Radians vs Degrees for angles  radians is equivalent to 180 ,  in degrees is equivalent to radians.180     For an acute angle  , i.e. 0 2  , we can use a right -angled triangle to define the three basic trigonometric functions: sine, cosine and tangent. opposite adjacent opposite ,, hypotenuse hypotenuse adjacent sin cos tan . O A O H H A         Mnemonics: Go ask your Grandma (Ah SOH) with the big foot (TOA CAH) for help! TOA-CAH-SOH For any angle  , we use a point P on the circumference of a circle of radius r centred at the origin O of the Cartesian place to define the three basic trigonometric functions. Let  be the angle through which the line OP has rotated anti-clockwise from the positive x-axis: sin , cos , tan . y r x r y x       Acronym for ASTC: All Science Teachers are Crazy, Add Sugar To Coffee… The positive acute angle between OP and the x-axis is known as the basic or reference angle, . When 0 2 , the following table gives the relationship between  and  for each quadrant that P lies in. Quadrant that P lies in Basic angle,  1st  2nd  3rd  4th 2 O A H  1st quadrant: ALL positive 2nd quadrant: SINE positive 3rd quadrant: TANGENT positive 4th quadrant: COSINE positive x  P(x, y) O x r y y ( 1, 1)P  y x 225  

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 3 of 11 2.1 Trigonometric Values of Special Angles Refer to Appendix A for methods to memorize and/or to derive the above results. 2.2 Graphs of Basic Trigonometric Functions (i) Graph of sin ,yx (ii) Graph of cos ,yx (iii) Graph of tan .yx §3 Inverse of Trigonometric Functions 1 1 1 sin sin . cos cos . tan tan . y x x y y x x y y x x y             Note that 1sin x denotes the inverse of sin x. It is NOT the reciprocal of sin x, i.e.     211 1 2 2 11sin sin , tan tansin tanx x x x xx       . Angle  degree 0o 30o 45o 60o 90o radian 0 6  4  3  2  sin 0 1 2 2 2 3 2 1 cos 1 3 2 2 2 1 2 0 tan 0 3 3 1 3 Not defined  x is continuous over .  y lies between –1 and 1 (inclusive).  It is 2 -periodic, i.e. sin( 2 ) sin .xx   Odd function, i.e.  sin sin .xx   Undefined at  2 1 , 2    x k k . (c.f. vertical asymptotes).  y can take any real values.  It is  -periodic, i.e. tan( ) tan .xx   Odd function, i.e.  tan tan .xx   x is continuous over .  y lies between –1 and 1 (inclusive).  It is 2 -periodic, i.e. cos( 2 ) cos .xx   Even function, i.e.  cos cos .xx 1 1 o45 o45 2 2 1 1 o30 o60 o60 3

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 4 of 11 3.1 Principal Values of Basic Trigonometric Functions From the graph o f sin ,yx we observe that several intervals of x correspond to the interval 11 y   (which are all the possible values of sin x ), such as 22 x   , 3 22 x , 35 ,22 x ... The interval 22 x   is chosen to be the principal values of x, or 1sin . y Example 3.1 List down a few values of x that satisfies 1sin . 2x What is the principal value of 1 1sin ? 2    Solution: 5 5 5..., 2 , 2 , , , 2 , 2 , ...6 6 6 6 6 6x                1 1Principal value of sin . 26    Note: Your calculator will only give the principal values of inverse trigonometric functions. The princip al values of the three basic trigonom etric functions are provided in MF -15. They are given as follows: 1 1 1 sin ( 1)22 0 cos ( 1) tan .22 xx xx x                 IMPORTANT: Principal value and basic angle are two different concepts. The next example illustrates this. Example 3.2 Solve 1sin , 2x where 0 2 .x  What is the principal value of 1 1sin ? 2   Solution: Basic angle = 1 1sin 2   .6  By “ASTC”, sin x is negative in the 3rd and 4th quadrants. Thus or 2 66 7 11or .66 x       Principal value of 1 1sin . 26    The modulus sign is absolutely necessary to find basic angle.

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 5 of 11 3.2 Relationships between Trigonometric Functions and Inverse Trigonometric Functions The general procedure to obtain expressions involving trigonometric functions of inverse trigonometric functions is to consider the geometry of an appropriate right-angled triangle (for acute angles). Example 3.3 Given that cos x  such that 0 90   , find tan in terms of x. Solution: This question is equivalent to simplifying   1tan cos . x Using the right-angled triangle, 21tan . x x  §4 Trigonometric Identities A trigonometric identity expressed in terms of x is an equality that involves trigonometric functions, and is true for all values of x (except possibly at singular points where the trigonometric function is not defined). The basic trigonometric identities are as follows: 1 RATIO IDENTITIES sintan , cos 0.cos AAA A coscot , sin 0.sin AAA A 2 RECIPROCAL IDENTITIES sec A = 1 cos A , cos 0.A cosec A = 1 sin A , sin 0.A cot A = 1 tan A , tan 0.A 3 NEGATIVE ANGLE IDENTITIES  sin sin .AA   cos cos .AA  tan tan .AA  4 PYTHAGOREAN IDENTITIES 22sin cos 1.AA 221 tan sec .AA 221 cot cosec .AA Explore: Solve Example 3.3