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Parametric equations: ⎧x = 2a sin 2 θ ⎪ 2a sin 3 θ ⎨ y = ⎪⎩ cos θ This is the curve described by a point P such that the distance OP = distance RS. , finding the side of a cube which has twice the volume of a given cube. Fig. 35. Polar equation: r = au Fig. 1. d = ( x 2 − x1 )2 + ( y2 − y1 )2 + (z 2 − z1 )2 z d γ P2 (x2, y2, z2) β P1 (x1, y1, z1) α y O x Fig. 2. 1 (see Fig. 10-1). 3. cos 2 α + cos 2 β + cos 2 γ = 1 or l 2 + m 2 + n 2 = 1 Direction Numbers Numbers L, M, N, which are proportional to the direction cosines l, m, n, are called direction numbers.

46. 47. 48. 49. 50. 51. 52. 69. 53. 57. 54. 58. 55. 59. 56. 60. 73. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. e. the angle whose sine is x or inverse sine of x is a many-valued function of x which is a collection of single-valued functions called branches. Similarly, the other inverse trigonometric functions are multiple-valued. For many purposes a particular branch is required. This is called the principal branch and the values for this branch are called principal values. TRIGONOMETRIC FUNCTIONS 50 Principal Values for Inverse Trigonometric Functions Principal values for x м 0 Principal values for x < 0 0 Ϲ sin–1 x Ϲ p/2 –p/2 Ϲ sin–1 x < 0 0 Ϲ cos–1 x Ϲ p/2 p/2 < cos–1 x Ϲ p 0 Ϲ tan–1 x < p/2 –p/2 < tan–1 x < 0 0 < cot–1 x Ϲ p/2 p/2 < cot–1 x < p 0 Ϲ sec–1 x < p/2 p/2 < sec–1 x Ϲ p 0 < csc–1 x Ϲ p/2 –p/2 Ϲ csc–1 x < 0 Relations Between Inverse Trigonometric Functions In all cases it is assumed that principal values are used.

For many purposes a particular branch is required. This is called the principal branch and the values for this branch are called principal values. TRIGONOMETRIC FUNCTIONS 50 Principal Values for Inverse Trigonometric Functions Principal values for x м 0 Principal values for x < 0 0 Ϲ sin–1 x Ϲ p/2 –p/2 Ϲ sin–1 x < 0 0 Ϲ cos–1 x Ϲ p/2 p/2 < cos–1 x Ϲ p 0 Ϲ tan–1 x < p/2 –p/2 < tan–1 x < 0 0 < cot–1 x Ϲ p/2 p/2 < cot–1 x < p 0 Ϲ sec–1 x < p/2 p/2 < sec–1 x Ϲ p 0 < csc–1 x Ϲ p/2 –p/2 Ϲ csc–1 x < 0 Relations Between Inverse Trigonometric Functions In all cases it is assumed that principal values are used.

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