Calculus: Analytic Geometry and Calculus, with Vectors

8.1 Trigonometric functions and their derivatives 445

9 Find the length L of the shortest ladder thatcan A

reach from level ground to a high wall when it must go over

a fence which is a feet high and b feet from the wall.

stns.: L = (aO + b3')34.

10 A heavy body is suspended from a rope, as in Figure

8.192, that runs up from the body to a pulley at B and

thence (a wonderful word) over two stationary pulleys at

A and B on the same horizontal level and back to B, where

%/

Figure 8.192

the pulley is tied to the other end of the rope. How should oldman gravity
select the 0 of the figure in order to gratify his desire to bring the heavy bodyto
its lowest possible position? Ans.: cos 0 = and 0 = a/3 or 0 = 60°.
11 If !P.1 is the length of a regular polygon ofn sides which is inscribed in a
circle of radius a, prove that jP,,1 = tan sin (ir/n) and hence that lim 1P = lira.

12 Supposing that 14 and B are constants not both 0 and that

y(x)=A'sin x+Bcosx,

calculate y(x) = y"(x) and show that

y"(x) _ -y(x)
In terms of the graph of y(x), tell precisely what it means to say that y"(x) < 0
when y(x) > 0 and y"(x) > 0 when y(x) < 0. Sketch graphs and verify your
conclusion when 11 = 0 and when B = 0. It is now required that we learn a
little trick that happens to be very important. Plot the point having coordi-
nates (A,B) and, as in Figure 8.193, let 4) be an angle having its terminal side on
the line running from the origin to the point. Let C = 112 + B2 and observe
that

so

and hence

11 = C cos 0 and B= C sin 0

y(x) = C(sin x cos ¢ + cos x sin ¢)

y(x) = C sin (x + 4,).

Finally, it is required that we learn some technical terminology by which this result

can be remembered. Functions of the form E sin (wt + 0) and E cos (cot + 0)

are called sinusoids (things like sines) of angular frequency co. Since x can be wt,

we have proved that the sum of two sinusoids having the same frequency co is also

a sinusoid having frequency w.

13 It is very easy to show that if k, d, B are constants for which k > 0 and

if

(1)

then

(2)

y = 11 sin kt + B cos kt

d2

dt2+k2y =0.

All we need to do is differentiate (1) twice and look at the result. The theory

of differential equations contains theorems which imply that if y is a function