Page 68 - mathsvol1ch1to3ans

68

1. Determine whether the following measurements produce one triangle, two triangles or no triangle:
◦
∠B = 88 , a = 23, b = 2. Solve if solution exists.
Solution:
◦
Let h = a sin B = 23 sin 88 = 23(0.99939082701) = 22.9859890214..
Since b < h triangle is not possible.
2. If the sides of a 4ABC are a = 4, b = 6 and c = 8, then show that 4 cos B + 3 cos C = 2.

Solution:
2
2
a + c − b 2 16 + 64 − 36
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cos B = = = 0.6875
2ac 64
2
2
a + b − c 2 16 + 36 − 64
cos C = = = −0.25
2ab 48
Now,4 cos B + 3 cos C = 4(0.6875) − 3(0.25) = 2
√ √
◦
3. In a 4ABC, if a = 3 − 1, b = 3 + 1 and C = 60 , find the other side and other two angles.
Solution:
√ 2 √ 2 √ √
1
2
2
2
c = a + b − 2ab cos C = 3 − 1 + 3 + 1 − 2 3 − 1 3 + 1
√ √ 2
= 3 + 1 − 2 3 + 3 + 1 + 2 3 − (3 − 1) = 6
. Hence c = 2.
√ √ √
a sin C 3 3 − 1 3 − 1
◦
sin A = = √ = √ = sin 15 .
c 2 6 2 2
◦
◦
B = 180 − (A + C) = 105 .
2
2
b + c − a 2
4. In any 4ABC, prove that the area 4 = .
4 cos A
Solution:
1
Area of the triangle = bc sin A
2
2bc sin A
=
4
2
2
b + c − a 2
= sin A
4 cos A
◦
5. In a 4ABC, if a = 12 cm, b = 8 cm and C = 30 , then show that its area is 24 sq.cm.
Solution:
1 1
◦
Area of the Triangle = ab sin C = (12)(8) sin 30 = 24 sq.cm.
2 2

6. In a 4ABC, if a = 18 cm, b = 24 cm and c = 30 cm, then show that its area is 216 sq.cm.

Solution:
p p
Area of the triangle = s(s − a)(s − b)(s − c) = 36(18)(12)(6) = 216 sq.cm.
7. Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at
the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the