58 CHAPTER 2. FUNCTIONS

In addition to this,

cos2x = cos2 x− sin2 x = 2cos2 x−1 = 1−2sin2 x (2.11)

Therefore, making use of the above identities,

cos(3x) = cos(2x+ x) = cos2xcosx− sin2xsinx

=(2cos2 x−1

)cosx−2cosxsin2 x

= 4cos3 x−3cosx (2.12)

For a systematic way to find cosine or sine of a multiple of x, see De Moivre’s theoremexplained in Problem 16 on Page 44.

Another very important theorem from Trigonometry is the law of cosines. Consider thefollowing picture of a triangle in which a,b and c are the lengths of the sides and A,B, andC denote the angles indicated.

A

B

C

a

b

c

The law of cosines is the following.

Theorem 2.3.6 Let ABC be a triangle as shown above. Then

c2 = a2 +b2 −2abcosC

Also, c ≤ a+b so the length of a side of a triangle is no more than the sum of the lengthsof the other two sides.

Proof: Situate the triangle so the vertex of the angle C, is on the point whose coordi-nates are (0,0) and so the side opposite the vertex B is on the positive x axis.

A

B

C

a

b

c

x

Then from the definition of the cosC, the coordinates of the vertex B are

(acosC,asinC)

while it is clear that the coordinates of A are (b,0). Therefore, from the distance formula,