# Elements of Plane and Spherical Trigonometry

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1899 edition. Excerpt: ...cos 0-. i sin 0 has n nth roots. This is equivalent to saying that every quantity has n nth roots, as is shown below. (1) Every quantity can be expressed in the form x.. yi. This will represent a real quantity if y = 0, a pure imaginary if x = 0, and a complex quantity if neither x nor y is 0. (2) x + yi = VxT+?(.--+ i, -V V V x' + y2 Vx2 + y2) If we put--x--cos 0 and-sin 0, which we Vx2 + y2 Vx2 + y2 ma. do, since the sum of the squares of these quantities is equal lo unity, we have x.. yi--V x1 + y2 (cos 0.f. i sin 0), where 0--tane x (3) By (112), taking the nth root of x.f. yi, we have fc+ji = + (cos + + sin + n n J Hence if we multiply the numerical value of V a;2 +?/2 by each of the n values of the second factor, we hall have the n, n values of y x.. yi. In an expression like the foregoing x + yi.= V x2.4. y (cos d.. i sin 0), the positive value of V x2.4-y2 is called the modulus of the complex quantity a;-f. and 0 is called the amplitude or argument of the same quantity. In the case of a real quantity the modulus is always equal to the quantity itself with the positive sign; while the amplitude is 0 if the quantity is positive, and-if it is negative. The modulus of a pure imaginary is the coefficient of i with the positive sign; and the amplitude is--if the coefficient of i is positive, and 3rr.-if the same coefficient is negative. EXERCISES. L Find the three cube roots of--8.--8 = 8 (-1 4. i.O) = 8 (cos 7T 4. i sin n), r_8 = f 8 (cos?_t + t sin 2hrJ if k = 0 =2 (cos 60 + isin 60") = 2(J + iv/3.i) if k = 1 = 2 (cos 180 + i sin 180) =--2 if& = 2--2 (cos 300 + i sin 300) = 2(J--ij/3.) 2. Find the square roots of 2 + 2/3.i. 2 + 2v/S.i = A(i + i V'3.i)=4 (cos.. + isin, 83. Exponential Values of cos o i sin 0. Expansion of sin o...