83. Graphical Methods. Any equation may be solved graphically by plotting the graphs of the two sides on the same pair of axes; then the points of intersection of the two curves, when projected on the x-axis, will determine the solutions. Example 1. Solve graphically the equation sin x = sin 2 x. The graphical solution is shown in Fig. 74. The solutions between 0° and 360° are a1 = 0°, α2 = 60°, α= 180°, α = 300°. 6. sin2 x cos x 7. √3 cos x + sin x = 9. 5 cos x - 2 sin x = = 2. 3 sin x cos x = 0. sin 2 x = 4 cos3 x. can be solved for x, if y is any number whatever between -1 and +1, and that there are an infinite number of solutions. Any one of these solutions is denoted by * Throughout this Chapter we shall suppose all angles measured in radians. Then (2) means that x is the number of radians in an angle (or arc†) whose sine is y; it is read "arc sine y” or an angle whose sine is y." 66 = The expressions y sin x, x = arcsin y, are two aspects of one relation, just as are the two statements "A is the uncle of B" and "B is the nephew of A"; either one implies the other; both mean the same thing. Likewise arccos y denotes an angle whose cosine is y; arctan y denotes an angle whose tangent is y; etc. = Whenever two quantities y and x are related in this dual manner, each is called the inverse of the other; thus, if y sin x, sin x is the inverse of arcsin y; and conversely, arcsin y is the inverse of sin a. Similarly, if y = cos x, cos x and arccos y are inverse to each other. An analogous notation is usual for tan x, and, indeed, for any function whatever. 85. Graphical Representation of Inverse Functions. equations (1) y = sin x and x = arcsin y Since the are equivalent, the same pairs of values of x and y which satisfy one of them satisfy the other. Hence either of these two equivalent equations is represented graphically by the curve drawn in §§ 55-56, p. 69. If we wish to study the arcsine function for its own sake, * The notation sin-1 y also is used very frequently to denote arcsin y; it is necessary to notice carefully that sin-1 y does not mean (sin y)−1. † If the unit angle is a radian, and the unit length be taken for the radius of a circle, the numerical measure of an angle at the center is equal to the numerical measure of the length of the intercepted arc. See § 44, p. 56. |