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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.

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The graphical solution is shown in Fig. 74. The solutions between 0° and 360° are a1 = 0°, α2 = 60°, α= 180°, α = 300°.

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6. sin2 x cos x
√2.

7. √3 cos x + sin x =
8. √3 sin x- COS X = √2.

9. 5 cos x - 2 sin x = = 2.
10. 6 cos 0 + 8 sin 0 = 9.
11. 5(1- sin x) = 2 cos x.
12. sin(60+x) = sin x.
13. sec2 x 4 sin2x = 0.
14. 4 sec2 x = 9 tan2 x.
15. ctn x + csc2 x = 3.
16. tan3 x = 3 tan x.
17. csc 3 x = csc 2 x.

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3 sin x cos x = 0.

sin 2 x = 4 cos3 x.

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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

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*

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

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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.

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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.

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it is convenient and customary to think of it as plotted in the ordinary manner, with the equation written in the form :

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FIG. 76

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