In passing to the limit as Ax approaches zero, we notice that

Au

Au approaches zero, and so also does

Then the limit of

2

the second factor on the right is unity, by (1) above. Passing

EXAMPLE 1. Uniform motion in a circle. If the position of a point P(x, y) at the time t, is given by the equations

x = a cos wt, y = a sin wt,

show that P moves in a circle, and find the magnitude and direction of the velocity and acceleration.

Squaring and adding the given equations, we get

x2 + y2 a2 cos2 wt + a2 sin2 wt = a2,

which is the equation of a circle (page 327), and the point P therefore moves on a circle.

wt with the x-axis. Since D0=w,

The radius OP makes an angle of ✪ the rate of change of 0, or the angular velocity of OP, is the constant w.

The table of values gives the times at which P crosses the axes in its first revolution.

Velocity of P. The components of the velocity parallel to the axes are (page 311)

α

P

As aw is constant, the point P moves uniformly in its path. This result may be verified as follows:

Since the angular velocity is w, in one second the radius OP will sweep over w radians, and hence the arc described by P will be aw (Theorem, page 171); that is, P will move a distance aw in one second.

The slope of the direction of the velocity is

Then the magnitude of the acceleration is

α √a2w1 cos2 wt + a2w4 sin2 wt aw2.

The slope of the direction of the acceleration is

In finding the derivative of sin u, use was made of the fact

1, and in finding this limit the angle was

measured in radians. Hence,

In applying the rules for differentiating and integrating trigonometric functions it must be remembered that the independent variable is measured in radians.

2. Find the derivatives of the following functions, by first expressing them in terms of the sine and cosine.

3. Find the derivative of each of the functions below, using the fundamental method of replacing x by x + Ax, etc.

(a) cos x. (b) tan x.

(c) cot x.

(d) sec x.

(e) csc x.

4. Find the maximum and minimum values of each of the trigonometric functions by the method of differentiation.

5. Differentiate the following functions, using (5) page 270.

(a) sin2 x. (b) cos2 2x. (c) sec2 x.

(d) 2 csc2 3x.

6. Find the coördinates of the points of inflection of the graph of

(a) sin x.

(b) cos x.

(c) tan x.

(d) cot x.

=

7. A point moves so that x 10 cos 2πt and y 10 sin 2πt. Show that the point moves in a circle, making one revolution per second. Find the components parallel to the axes and the magnitude and the direction of the velocity and acceleration. Where does the point start?

8. As in the preceding exercise, discuss the motion of a point if x 10 cos (2πt + π/2) and y = 10 sin (2πt + π/2). Where does the point start? 9. Discuss the motion of a point if x = 4 cos (πx/2 + π/3), y 4 sin (πx/2 + π/3).

10. The position of a point P moving on a circle of radius 5 so that OP rotates through 2 radians per second is given by

x = 5 cos 2t, y 5 sin 2t.

If M is the projection of P on the x-axis (see figure for Example 1 of the preceding section), find the position (s = OM), velocity, and acceleration of M at any time. Find when and where the values of s, v, and a are greatest and least.

11. A chip on the surface of a pond moves up and down with the waves according to the law s = sint. Find the velocity and acceleration. Determine the maximum and minimum values of s, v, and a. Plot the three graphs on the same axes.

12. Find the area of an arch of each of the curves, (a) sin x. (b) sin x. (c) cos 2x. (d) 3 sin x.

A

13. In a right triangle ABC, it was found that b 10 inches, and 45°. Find a and c, and the error in each due to an error of 0°.1 in A. 14. In a triangle a = 2, b = 4, C 60°. Find c by the law of cosines

=

and the error in c due to an error of 0°.2 in C.

15. Given sin 30° = 1 .5000 and cos 30° = √3/2 = .8660, find approximate values of sin 30°.1 and cos 30°.1.

= cos 45° = √2/2 = .7071, find approximate values of

16. Given sin 45° sin 45°.1 and cos 45°.1.

17. Given sin 60° = √3/2 = .866 and cos 60° 1242 .500, compute a three-place table of sines and cosines of 60°.1, 60°.2, 60°.3, 60°.4, 60°.5. 18. A ship is anchored 300 yards from a straight shore, along which a searchlight is played. If the light is turned uniformly at the rate of one radian per minute, how fast will the beam of light be moving along the shore when it makes an angle of 30° with the shore?

-19. A balloon is ascending vertically at the rate of 5 feet per second. An observer stands 200 feet from the point at which the balloon started. How fast is the angle of elevation changing when it equals 60°?

20. A level road approaches a hill 300 feet high. An observer on the hill notes that the angle of depression of a motorcycle is 80° and that the angle decreases 1° in three seconds. Find approximately the speed of the motorcycle.

129. Graph of the Function a sin (bx + c). Harmonic Curves. Suppose at first that c = 0, and consider the special

The graph of this function may be obtained from that of sin x by first dividing the abscissas by b, which gives the graph

of sin bx [(4), page 152], and then multiplying the ordinates of points on the graph of sin bx by a [(3), page 152]. Since the graph of sin x repeats itself in intervals of 2π, the graph of (1) will repeat in intervals of 2π/b, so that the function