103 Trigonometry Problems

1. Trigonometric Fundamentals 47
It follows that
(v − u) · (v − u) = u · u + v · v − 2|u||v| cos θ,
or v · v − 2u · v + u · u = v · v + u · u − 2|u||v| cos θ. Hence
cos θ = u · v
|u||v| .
Cauchy–Schwarz Inequality
Let u =[a,b] and v =[m, n], and let θ be the angle formed by the two vectors
when they are placed tail to tail. Because | cos θ| ≤1, by the previous discussions,
we conclude that (u · v) 2 ≤ (|u||v|) 2 ; that is,
(
(am + bn) 2 ≤ a 2 + b 2)( m 2 + n 2) .
Equality holds if and only if | cos θ| =1, that is, if the two vectors are parallel. In
any case, the equality holds if and only if u = k · v for some nonzero real constant
k; that is,
m a = n b = k.
We can generalize the definitions of vectors for higher dimensions, and define
the dot product and the length of the vectors accordingly. This results in Cauchy–
Schwarz inequality: For any real numbers a 1 ,a 2 ,...,a n , and b 1 ,b 2 ,...,b n ,
(
)(
)
a1 2 + a2 2 +···+a2 n b1 2 + b2 2 +···+b2 n ≥ (a 1 b 1 + a 2 b 2 +···+a n b n ) 2 .
Equality holds if and only if a i and b i are proportional, i = 1, 2,...,n.
Now we revisit Example 1.11. Setting n = 3, (a 1 ,a 2 ,a 3 ) = (√ x, √ y, √ z ) ( )
,
(b 1 ,b 2 ,b 3 ) = √x 1
, √ 2
y
, √ 3 z
in Cauchy–Schwarz inequality, we have
1
x + 4 y + 9 ( 1
z = (x + y + z) x + 4 y + 9 )
≥ (1 + 2 + 3) 2 = 36.
z
Equality holds if and only if x 1 = y 2 = z 3 ,or(x,y,z)= ( 1
6
, 1 3 , 1 2)
.
Radians and an Important Limit
When a point moves along the unit circle from A = (1, 0) to B = (0, −1), ithas
traveled a distance of π and through an angle of 180 ◦ . We can use the arc length as