The Art and Craft of Problem Solving

2.4 OTHER IMPORTANT STRATEGIES 57

Now we introduce a simple trig tool: Given an expression of the form acos 0 + bsin O,
write

acos O+bsino =va^2 +b^2 (

2

a

2

cosO+

2

b

2

sin o).

va +b va +b

This is useful, because

a

and

b

are respectively the sine and cosine

va 2 + b 2 va 2 + b 2

of the angle a:= arctan(a/b).

Consequently,

a

acos 0 + bsinO = Va^2 +b^2 (sinacos 0 + cos asin 0)

= Va^2 +b^2 sin(a+O).

In particular, we have

sinx+cosx = v'2sin (x+ �).

Applying this, equation (9) becomes (note that V6^2 + 8^2 = 10 )

5 v'2v'2 sin ( � + �) = 10 (� cos 0 + � sin 0 ).

Hence, if a = arctan(3/4), we have

sin (�+�) = sin(a+ 0).

Equating angles yields 0 = 1r/2 - 2a. Thus

x= cosO = sin(2a) = 2sinacosa = 2 (�) (�) = ��. •

Converting a problem to geometric or pictorial forms usually helps, but in some

cases the reverse is true. The classic example, of course, is analytic geometry, which

converts pictures into algebra. Here is a more exotic example: a problem that is geo­

metric on the surface, but not at its core.

Example 2.4.5 We are given n planets in space, where n is a positive integer. Each

planet is a perfect sphere and all planets have the same radius R. Call a point on the

surface of a planet private if it cannot be seen from any other planet. (Ignore things

such as the height of people on the planet, clouds, perspective, etc. Also, assume that