The Chemistry Maths Book, Second Edition

70 Chapter 3Transcendental functions 0 Exercises 9, 10 Periodic functions A functionf(x)with the property f(x 1 ± 1 a) 1 = 1 f(x ...

3.2 Trigonometric functions 71 EXAMPLE 3.7A harmonic wave travelling in the positive xdirection (a plane wave) is described by t ...

72 Chapter 3Transcendental functions This is one of the simplest second-order differential equations (Chapter 12) and a solution ...

3.4 Trigonometric relations 73 To overcome this ambiguity, a principal valuehas been defined for each inverse function: (3.18) T ...

74 Chapter 3Transcendental functions so thath 1 = 1 csin A 1 = 1 asin C. Then Similarly for the third angle and side. For the co ...

3.4 Trigonometric relations 75 Also, puttingx 1 = 1 y, sin 12 x 1 = 121 sin 1 x 1 cos 1 x (3.23) cos 12 x 1 = 1 cos 2 x 1 − 1 si ...

76 Chapter 3Transcendental functions EXAMPLE 3.14Express and in terms ofsin 1 θandcos 1 θ. From equations (3.21), Figure 3.11 sh ...

3.5 Polar coordinates 77 Making use of the expressions (3.21) for the sines of the sum and difference of angles An important spe ...

78 Chapter 3Transcendental functions The figure shows that the position of the point P can be specified not only by its cartesia ...

3.5 Polar coordinates 79 converting from cartesian coordinates to polar coordinates. The correct value of θis determined by the ...

80 Chapter 3Transcendental functions 3.6 The exponential function An exponential function has the form f(x) 1 = 1 a x (3.29) in ...

3.6 The exponential function 81 The terms are decreasing rapidly in magnitude, and the first five are sufficient (compare Exampl ...

82 Chapter 3Transcendental functions with the 1 + 1 sign for growth and the 1 − 1 sign for decay. The proportionality factor kis ...

3.7 The logarithmic function 83 3.7 The logarithmic function The logarithmic function 6 is the inverse function of the exponenti ...

84 Chapter 3Transcendental functions The graphs of all the logarithmic functions are similar, with properties log 111 = 1 0, log ...

3.7 The logarithmic function 85 0 Exercises 39 EXAMPLE 3.27What notto do. A surprisingly common error is to put ln 1 (x 1 + 1 y) ...

86 Chapter 3Transcendental functions The pH is an example of the use of the logarithm as a scale of measure. Other examples of l ...

The following conclusions can be drawn from the table. (a)ln 1 xvaries slowly compared to x. In fact it varies more slowly than ...

88 Chapter 3Transcendental functions The hyperbolic cosine and sine are defined in terms of the exponential function as (3.47) ( ...

3.10 Exercises 89 The relation forcosh − 1 xshows that there exist two values of the function for each value ofx(> 1 1). Thes ...