Pure mathematics

Then, since, Art. 15, the sine of an angle is the same as the sine of its supplement, there are two values of A which

satisfy the equality, sin A = sin B, and these values are

supplementary.

Let A, A' be the two values, then the relation between them is A + A'

180°.

If a is not greater than b, then A is not greater than B, and there is no doubt as to which value of A is to be taken. If, however, a is greater than b—that is, if the given angle is opposite to the less of the given sides, we must have A greater than B, and both values of A may satisfy this condition. This particular case, when the given angle is opposite to the less of the given sides, is called the ambiguous case. We will illustrate this geometrically.

42. The ambiguous case.

Let a, b, B be given to construct the triangle.
Draw the line BC equal to the

given side a, and draw BA making
an angle B with BC.

Then with centre C and radius
CA equal to b describe an arc
meeting BA or BA produced in
A and A', and join CA and CA'.
Each of the triangles ABC,
ABC satisfies the given conditions. B4
For, in the triangle ABC, we have BC
ZABC

a, AC = b, and

B; and in the triangle A'BC, we have BC = a, AC b, and Z A'BC

Again, the sides BA and BA' correspond to the two value

of c which are obtained from the two values of A in the

And the angles ACB and A'CB correspond to the two values of C which would also be found.

COR. If a perpendicular CD be drawn from C upon AA', and if c' and c be the lengths of BA' and BA respectively, it may be easily shown that c' + c = 2 a cos B, and c' ~ c = 2 b cos A.

Ex. VIII.

Solve the following triangles, having given—

1. b 12, c = 6, A = 60°.

8. a = Log 237 Log 341

237, b

9.6772640,

2.3747483, L sin 28° 24′

2.5327544, L sin 19° 18' 9-5191904,

L sin 19° 19' = 9.5195510.

9. C

26° 32', and ab:: 3 : 5, find A, B.

Log 23010300, L cot 13° 16' 10-6275008,

L tan 46° 40′ = 10·0252805, L tan 46° 41' = 10·0255336.

L tan 29° 12′ = 9.7473194, L tan 29° 13' =

9.6506199,

9.7476160.

11. a

3, b

= 2 A

60°, find B, C, and c. Log 2 3010300, log 3 = 4771213,

9.7614638,

L sin 35° 15′ = 9.7612851, L sin 35° 16′
Log 1.37971397847, log 1.3798 = 1398161.

13. If c, c' be the two values of the third side in the ambiguous case when a, b, A are given, show that—

14. If a, b, A are given, show from the equation

that if c and c' be the two values of the third side

15. Show also from the same equation that there is no ambiguous case when a = b sin A, and that c is impossible when a < b sin A.

16. Having a 6, A, B, solve the triangle.

17. Given the ratios of the sides, and the angle A, solve the triangle.

18. If in a triangle tan (B-C) = tan2 cot, show that b cos & = c.

CHAPTER X.

HEIGHTS AND DISTANCES.

43. We shall now show how the principles of the previous chapters are practically applied in determining heights and distances.

We have not space here to describe the instruments by which angles are practically measured, but we shall assume that they can be measured to almost any degree of accuracy.

44. To find the height of an accessible object.

Let AC be the object, and let A any distance BC from its foot be measured.

At B let the angle of elevation
ABC be observed.

Suppose BC = a, ▲ ABC
Then, we have

= 0.