Calculus: Analytic Geometry and Calculus, with Vectors

444 Trigonometric functions

y = sin x to obtain a procedure for sketching the graph of y = 3 sin x, and sketch
the graph.
3 Equate the derivatives of the two members of the identity in the left
column and show that obvious simplifications give the identity in the right column
when

(a) sin 2x = 2 sin x cos x cos 2x = cost x - sine x

(b) cos 2x = cos2 x - sin2 x sin 2x = 2 sin x cos x

(c) sin2 x + cos' x= 1 0 = 0

(d) sin (x + 4,)

= sin x cos 0 + cos x sin 4,

cos (x + 4>)

= cos x cos 4, - sin x sin ¢

(e) cos (x + 4,)

= cos x cos 4> - sin x sin 4>

sin (x + ¢)

= sin x cos 4) + cos x sin 40

(f) cos'x=1+2s2x 2sinxcosx=sin2x

(g) sin' x=1 - cos 2x 2 2 sin x cos x = sin 2x

(h) sin X+1 = cos x cos(x+?}= -sin x

(i) exez = e2z

(j) log x' = 2 log x

4 Supposing that 0 S x < ir/2, let f(x) = tan x and show that

f' (x) = sec' x

f"(x) = 2 sec' x tan zcl

f"'(x) = 2 sec' x+ 4 sec' x tan' x,

and obtain the next derivative. Show that each derivative of higher order will

also be a sum of terms of the form .4 sec-" x tang x, where 14, p, q are non-

negative integers, and hence that fi')(x) >-- 0 for each it and x.

5 Supposing that 0 < x < a/2, calculate the first three derivatives of sec x.

Show that these and all derivatives of higher order are nonnegative.

6 Again supposing that 0 < x < it/2, calculate some derivatives of \/tan x

and try to decide whether they are all nonnegative.

7 Prove the formula

sing

fl + sin 8d0= sec0-tang+0+c

by differentiating the right side.

8 The length L of the longest beam that can be taken in a horizontal position

Figure 8.191 around the comer of Figure 8.191 is the length L of the

shortest line segment placed like the one in the figure.

Find L. Ans.: L = (a%* + b%i)3i. Remark: Putting the

equation in the form a'% + h% = L% can make us

wonder why (a,b) should, for a fixed L, be a point on a

hypocycloid.