Basic Engineering Mathematics, Fifth Edition

206 Basic Engineering Mathematics

or (b)

1

2

absinCor

1

2

acsinBor

1

2

bcsinA

or (c)

√

[s(s−a)(s−b)(s−c)]wheres=

a+b+c

2

23.3 Worked problems on the solution

of triangles and their areas

Problem 1. In a triangleXYZ,∠X= 51 ◦,

∠Y= 67 ◦andYZ= 15 .2cm. Solve the triangle and

find its area

The triangleXYZis shown in Figure 23.2. Solving the

triangle means finding∠Zand sidesXZandXY.

X

Y x 5 15.2cm Z

z y

518

678

Figure 23.2

Since the angles in a triangle add up to 180◦,

Z= 180 ◦− 51 ◦− 67 ◦= 62 ◦

Applying the sine rule,

15. 2

sin51◦

=

y

sin67◦

=

z

sin62◦

Using

15. 2

sin51◦

=

y

sin67◦

and transposing gives y=

15 .2sin67◦

sin51◦

= 18 .00cm=XZ

Using

15. 2

sin51◦

=

z

sin62◦

and transposing gives z=

15 .2sin62◦

sin51◦

= 17 .27cm=XY

Area of triangleXYZ=

1

2

xysinZ

=

1

2

( 15. 2 )( 18. 00 )sin 62◦= 120 .8cm^2

(or area=

1

2

xzsinY=

1

2

( 15. 2 )( 17. 27 )sin67◦

= 120 .8cm^2 )

It is always worth checking with triangle problems that

the longest side is opposite the largest angle and vice-

versa. In this problem,Yis the largest angle andXZis

the longest of the three sides.

Problem 2. Solve the triangleABCgiven

B= 78 ◦ 51 ′,AC= 22 .31mm andAB= 17 .92 mm.

Also find its area

TriangleABCis shown in Figure 23.3. Solving the

triangle means finding anglesAandCand sideBC.

A

Ba C

b 5 22.31mm

c^5

17.92mm

788519

Figure 23.3

Applying the sine rule,

22. 31

sin78◦ 51 ′

=

17. 92

sinC

from whichsinC=

17 .92sin78◦ 51 ′

22. 31

= 0. 7881

Hence, C=sin−^10. 7881 = 52 ◦ 0 ′or 128◦ 0 ′

SinceB= 78 ◦ 51 ′,Ccannot be 128◦ 0 ′, since 128◦ 0 ′+

78 ◦ 51 ′is greater than 180◦. Thus, onlyC= 52 ◦ 0 ′is

valid.

AngleA= 180 ◦− 78 ◦ 51 ′− 52 ◦ 0 ′= 49 ◦ 9 ′.

Applying the sine rule, a

sin49◦ 9 ′

=

22. 31

sin78◦ 51 ′

from which a=^22 .31sin49

◦ 9 ′

sin78◦ 51 ′

= 17 .20mm

Hence,A= 49 ◦ 9 ′,C= 52 ◦ 0 ′andBC= 17 .20mm.

Area of triangleABC=

1

2

acsinB

=

1

2

( 17. 20 )( 17. 92 )sin78◦ 51 ′

= 151 .2mm^2

Problem 3. Solve the trianglePQRand find its

area given thatQR= 36 .5mm,PR= 29 .6mmand

∠Q= 36 ◦