Higher Engineering Mathematics

INTRODUCTION TO TRIGONOMETRY 127

B

Applying the sine rule:

32. 91

sin 64◦

=

25. 0

sinF

from which, sinF=

25 .0 sin 64◦

32. 91

= 0. 6828

Thus ∠F=sin−^10. 6828

= 43 ◦ 4 ′or 136◦ 56 ′

F= 136 ◦ 56 ′is not possible in this case since
136 ◦ 56 ′+ 64 ◦is greater than 180◦. Thus only
F= 43 ◦ 4 ′is valid

∠D= 180 ◦− 64 ◦− 43 ◦ 4 ′= 72 ◦ 56 ′

Area of triangleDEF=^12 dfsinE

=^12 (35.0)(25.0) sin 64◦=393.2 mm^2.

Problem 24. A triangleABChas sidesa=

9.0 cm, b= 7 .5 cm andc= 6 .5 cm. Determine

its three angles and its area.

TriangleABCis shown in Fig. 12.24. It is usual first
to calculate the largest angle to determine whether
the triangle is acute or obtuse. In this case the largest
angle isA(i.e. opposite the longest side).

Applying the cosine rule:

a^2 =b^2 +c^2 − 2 bccosA

from which, 2bccosA=b^2 +c^2 −a^2

and cosA=

b^2 +c^2 −a^2

2 bc

=

7. 52 + 6. 52 − 9. 02

2(7.5)(6.5)

= 0. 1795

Figure 12.24

Hence A=cos−^10. 1795 = 79 ◦ 40 ′ (or 280◦ 20 ′,
which is obviously impossible). The triangle is thus
acute angled since cosAis positive. (If cosAhad
been negative, angleA would be obtuse, i.e. lie
between 90◦and 180◦).

Applying the sine rule:

9. 0

sin 79◦ 40 ′

=

7. 5

sinB

from which,

sinB=

7 .5 sin 79◦ 40 ′

9. 0

= 0. 8198

Hence B=sin−^10. 8198 = 55 ◦ 4 ′

and C= 180 ◦− 79 ◦ 40 ′− 55 ◦ 4 ′= 45 ◦ 16 ′

Area=

√

[s(s−a)(s−b)(s−c)],

where s=

a+b+c

2

=

9. 0 + 7. 5 + 6. 5

2

= 11 .5cm

Hencearea

=

√

[11.5(11. 5 − 9 .0)(11. 5 − 7 .5)(11. 5 − 6 .5)]

=

√

[11.5(2.5)(4.0)(5.0)]=23.98 cm^2

Alternatively, area=^12 absinC

=^12 (9.0)(7.5) sin 45◦ 16 ′=23.98 cm^2.

Now try the following exercise.

Exercise 58 Further problems on solving

triangles and finding their areas

In Problems 1 and 2, use the cosine and sine

rules to solve the trianglesPQRand find their

areas.

1.q=12 cm,r=16 cm,P= 54 ◦

[

p= 13 .2 cm,Q= 47 ◦ 21 ′,

R= 78 ◦ 39 ′, area= 77 .7cm^2

]

2.q= 3 .25 m,r= 4 .42 m,P= 105 ◦

[

p= 6 .127 m,Q= 30 ◦ 50 ′,

R= 44 ◦ 10 ′, area= 6 .938 m^2

]

In problems 3 and 4, use the cosine and sine

rules to solve the trianglesXYZand find their

areas.

3.x= 10 .0 cm,y= 8 .0 cm,z= 7 .0cm

[

X= 83 ◦ 20 ′,Y= 52 ◦ 37 ′,

Z= 44 ◦ 3 ′, area= 27 .8cm^2

]