The Chemistry Maths Book, Second Edition

74 Chapter 3Transcendental functions

so thath 1 = 1 csin A 1 = 1 asin C. Then

Similarly for the third angle and side.

For the cosine rule, from Figure 3.15 and Pythagoras’ theorem,

a

2

1 = 1 h

2

1 + 1 (b 1 − 1 x)

2

1 = 1 h

2

1 + 1 b

2

1 + 1 x

2

1 − 12 bx, c

2

1 = 1 h

2

1 + 1 x

2

and the rule follows sincex 1 = 1 c 1 cos 1 A.

EXAMPLE 3.10Given the lengthsa 1 = 1 2, c 1 = 13 , and the angleB 1 = 1 π 23 of the triangle

in Figure 3.14, find the third side and the other angles.

Given two sides and the included angle, we use the cosine rule to find b:

b

2

1 = 1 a

2

1 + 1 c

2

1 − 12 accos 1 B 1 = 141 + 191 − 1 12 cos 1 π 23

andcos 1 π 231 = 1122. Thereforeb

2

1 = 141 + 191 − 161 = 17 and

To find the other two angles we use the cosine rule for one; for example,

a

2

1 = 1 b

2

1 + 1 c

2

1 − 12 bccos 1 A

so that

cos 1 A 1 = 1 (b

2

1 + 1 c

2

1 − 1 a

2

) 22 bc 1

Then C 1 ≈ 1 180° 1 − 1 60° 1 − 1 40.89° 1 = 1 79.11°.

0 Exercises 14–17

Compound-angle identities

The sines and cosines of the sum and difference of two angles are

sin(x 1 + 1 y) 1 = 1 sin 1 x 1 cos 1 y 1 + 1 cos 1 x 1 sin 1 y

sin(x 1 − 1 y) 1 = 1 sin 1 x 1 cos 1 y 1 − 1 cos 1 x 1 sin 1 y

(3.21)

cos(x 1 + 1 y) 1 = 1 cos 1 x 1 cos 1 y 1 − 1 sin 1 x 1 sin 1 y

cos(x 1 − 1 y) 1 = 1 cos 1 x 1 cos 1 y 1 + 1 sin 1 x 1 sin 1 y

(3.22)

==











≈

−

27

2

7

40 89

1

and A cos. °

b= 7.

sin sin AC

ac

=