From (1), L sin 4 = 1102 + 1⁄2 110b + 1⁄2 110c − 110 (b + c) + L cos

= (110b+ 110€) + 1102 + L cos

(2), 110α = 110 (b + c) + L cos & − 10;

which give and a respectively.

A

87. Let two sides be given and an angle opposite to one of them. (a, b, A.)

But if a

We have shewn in Art. 72, that with these data the solution is ambiguous, unless a be greater than b. be greater than b, we then have

sin A, where B is an angle less than 90°.

If A be nearly 90°, the first formula will not give the value of B very exactly, because the increment of sin A does not in that case vary as the increment of A, and it is also very small; App. 11. 11;-in this case any one of the last three forms may be used. And the second or the

A

2

Ꭺ

third form must be taken, according as COS or sin is

2

A

the greater, i. e. as

is less or greater than 45o.

Art. 64.

2

2

The fourth form is applicable in all cases except where nearly 90o. Art. 63.

89. EXAMPLES. 1. If BC be a perpendicular object standing on a horizontal plane, its height may be determined by measuring in that plane a line AC, which is called a base, and observing the angle BAC with a proper instrument.

For BC AC. tan BAC;

=

.. 110 BC1104C + L tan BAC - 10.

A

2. If it be not possible to come to the foot of the object, let a base AD be measured, such that the points D, A, C may be in the same line, and let the angles BDA, BAC be observed.

Here we have two angles given and a side of the ABAD. By determining the side BA we can, from the right-angled triangle BAC, find the height of BC.

B

B

sin (BAC - BDA)

.. 110BC=1104D+LsinBDA+Lsin BAC-Lsin(BAC-BDA) – 10.

-

3.

If D be not in the line AC, the height BC can yet be determined by taking proper observations.

At A let the angles BAC and BAD

be observed, and at D the angle BDA.

Then in the ABDA we have the D

angles BDA, BAD and the side AD given. If then BA be determined from

these data, we can find BC from the right-angled triangle BAC.

B

110BC= \104D+ L sin BAC + L sin BDA–L sin (BDA+BAD)−10.

It is evident that this determination of CB is not affected by the circumstance of D lying out of the horizontal plane which passes through A and C. Hence it follows, that if a straight base AD be measured in any direction from A, and the angles BAC, BAD, BDA be observed, we shall be provided with sufficient data for finding the height of B above the horizontal plane passing through A.

4. Required to find the breadth of a river AD, from observations made from the top of a tower BC whose height is known. (Figure to Ex. 2.)

At B let the angles of depression of the points D and A below a horizontal line passing through B be observed. Since such a line will be parallel to CD, BDC and BAC are equal to these angles of depression.

K

By working a few examples of this kind the student will soon become familiar with the management of Trigonometrical formulæ. We add one more Example a little more difficult than the preceding, before leaving this part of our subject.

5. Required the error in height arising from a small given error in an observation of the angle in Example 1.

Let BC=h, AC = a, 4 BAC = A.

Let Sh be the error in height produced by an error SA of the observed angle.

B

sin & A

(cos A)2

cos (4 +84). cos A

sin {(4 +84) - A}

cos (4 +84). cos A

Since cos A = cos (4 + dA), nearly. Art. 61.

COR. Hence we may determine when the error in height, arising from a small given error in the observed angle, will

Now h being constant, and 84 being given, this expression, which is the error in altitude, will be the least when sin 2A is the greatest, that is, when 24 = 90°, or A=45°.

The observer therefore ought to move along AC till ▲ BAC = 45o, and then by measuring AC he will determine CB, which is in this case equal to AC, with the least chance. of error.

90. To find the area of a triangle, the sides being given.

Let ABC be the triangle, and from C draw CD perpendicular to AB, or to AB produced either way.

Then Area of triangle ABC, being half the rectangle on the same base and between the same parallels,

= 1⁄2 AB . CD = ↓ AB. AC . sin CAB

√ {S. (S−a). (S−b). (S−c)}. Art. 76. Cor. 1.