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     