Cambridge International Mathematics

Example 6 Self Tutor

Find, correct to 1 decimal place, the measure of angleCin triangle ABC if AC=8cm,

AB=12cm, and angleBmeasures 28 o.

sinC

c

=

sinB

b

fsine ruleg

)

sinC

12

=

sin 28o

8

) sinC=

12 £sin 28o

8

Now sin¡^1

μ

12 £sin 28o

8

¶

¼ 44 : 8 o

and since the angle at C could be acute or obtuse,

) C¼ 44 : 8 o or (180¡ 44 :8)o

) Cmeasures 44 : 8 oif it is acute, or 135 : 2 oif it is obtuse.

In this case there is insufficient information to determine the actual shape of the triangle.

The validity of the two answers in the above example can be demonstrated by a simple construction.

Step 1: Draw AB of length 12 cm and

construct an angle of 28 oat B.

Step 2: From A, draw an arc of radius 8 cm.

Sometimes there is information given in the question which enables us torejectone of the answers.

Example 7 Self Tutor

Find the measure of angleLin triangle KLM given that LKM measuresb 52 o,

LM= 158m, and KM= 128m.

By the sine rule,

sinL

128

=

sin 52o

158

) sinL=

128 £sin 52o

158

Now sin¡^1

μ

128 £sin 52o

158

¶

¼ 39 : 7 o

) L¼ 39 : 7 o or (180¡ 39 :7)o¼ 140 : 3 o

But KM<LM, so we know angleL<angleK. Hence L¼ 39 : 7 o.

8cm

8cm

12 cm

28°

135.2°

44.8°

AB

Cz

Cx

12 cm

AB28°

12 cm

8cm

28°

AB

C

K M

L

128 m

158 m

52°

Further trigonometry (Chapter 29) 587

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Y:\HAESE\IGCSE01\IG01_29\587IGCSE01_29.CDR Monday, 27 October 2008 2:52:49 PM PETER