307. Circular functions of complex angles. When a is a complex quantity, the functions sin x and cos x have at present no meaning. For real values of x we have already shewn in Arts. Let us define sin x and cos x, when x is complex, so that these relations may always be true, i.e. for all values of a let When is complex, the quantities sine and cos x are then only short ways of writing the series on the right-hand sides of (1) and (2). 308. We have then, for all values of x, real or complex, Hence for all values of x, real or complex, we have 309. We can now shew that the Addition and Subtraction Theorems hold for imaginary angles, i.e. that, whether a be real or complex, then 310. It follows that all formula which have been proved for real angles and which are founded on the Addition and Subtraction Theorems are also true when we substitute for the real angle any complex quantity. For example, since cos 30 = 4 cos3 0 – 3 cos 0, where is real, it follows that cos 3 (x + yi) = 4 cos3 (x + yi) — 3 cos (x + yi). Again, since, by De Moivre's Theorem, we know that cos no + i sin n✪ is always one of the values of (cos + i sin )n, when is real and n has any value, it follows that cos n (x + yi) + i sin n (x + yi) is always one of the values of [cos (x+yi) + i sin (x + yi)]”. In 311. Periods of complex circular functions. equations (1) and (2) of Art. 309 let a be complex and let y = 2π. Then sin (x+2)= sin x cos 2π + cos x sin 2π = sin x, and cos (x+2π) = cos x cos 2π- sin x sin 2π = COS X. Hence sin x and cos x both remain the same when x is increased by 2π. Similarly they will remain the same when a is increased by Hence, when x is complex, the expressions sin x and cos a are periodic functions whose period is 27. This corresponds with the results we have already found for real angles. (Art. 61.) sin^-1 10. {sin (a−0)+e±ai sin 0}”—sin”—1 a {sin (a --n0)+e±ai sin no}. 312. In the formulæ of Art. 308 if x be a pure imaginary quantity and equal to yi, we have, since whether y be real or complex, is called the hyperbolic sine of y and is written sinh y. L. T. 24 is called the hyperbolic cosine of y and is written cosh y. [It will be observed that the values of sinhy and cosh y are obtained from the exponential expressions for siny and cosy by simply omitting the i's.] The hyperbolic tangent, secant, cosecant, and cotangent are obtained from the hyperbolic sine and cosine just as the ordinary tangent, secant, cosecant, and cotangent are obtained from the ordinary sine and cosine. The hyperbolic cosine and sine have the same relation to the curve called the rectangular hyperbola that the ordinary circular cosine and sine have to the circle. Hence the use of the word hyperbolic. and So 314. From Arts. 312 and 313 we clearly have cos (yi) = coshy, sin (gi)= i sinh y. |