MISCELLANEOUS EXAMPLES. LXI.

(1) A balloon is at a height of 2500 feet above a plain and its angle of elevation at a point in the plain is 40° 35'. How far is the balloon from the point of observation?

(2) A tower standing on a horizontal plain subtends an angle of 37° 19′ 30′′ at a point in the plain distant 369.5 feet from the foot of the tower. Find the height of the tower.

(3) The shadow of a tower on a horizontal plain in the sunlight is observed to be 176·23 feet and the elevation of the sun at that moment is 33° 12'. Find the height of the tower.

(4) From the top of a tower 163·5 feet high by the side of a river the angle of depression of a post on the opposite bank of the river is 29° 47′ 18′′. Find the distance of the post from the foot of the tower.

(5) Given a=673·12, b=415·89 chains, C=90°, find A and B. (6) Given a=576·12, c=873·14 chains, C'=90°, find 6 and 4.

(7) From the top of a light-house 112.5 feet high, the angles of depression of two ships, when the line joining the ships points to the foot of the light-house, are 27° 18′ and 20° 36' respectively. Find the distance between the ships.

(8) From the top of a cliff the angles of depression of the top and bottom of a light-house 97.25 feet high are observed to be 23° 17′ and 24° 19′ respectively. How much higher is the cliff than the light-house?

(9) Find the distance in space travelled in an hour, in consequence of the earth's rotation, by St Paul's cathedral. (Latitude of London=51° 25′, earth's diameter=7914 miles.)

(10) The angle of elevation of a balloon from a station due south of it is 47° 18′ 30′′, and from another station due west of the former and distant 671 38 feet from it the elevation is 41° 14'. Find the height of the balloon.

CHAPTER XVI.

ON THE RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE.

231. The three sides and the three angles of any triangle, are called its six parts.

By the letters A, B, C we shall indicate

geometrically, the three angular points of the triangle ABC ; algebraically, the three angles at those angular points respectively.

By the letters a, b, c we shall indicate the measures of the sides BC, CA, AB opposite the angles A, B, C respectively.

232. I. We know that, A+B+C=180o. [Euc. 1. 32.] 233. Also if A be an angle of a triangle, then A may have any value between 0° and 180°. Hence,

(i) sin A must be positive (and less than 1),

(ii) cos A may be positive or negative (but must be numerically less than 1),

(iii) tan A may have any value whatever, positive or negative.

234. Also, if we are given the value of

(i) sin A, there are two angles, each less than 180°, which have the given positive value for their sine.

(ii) cos A, or (iii) tan A, then there is only one value of A, which value can be found from the Tables.

known, when the value of any one of its Ratios is given. Similar remarks of course apply to the angles B and C.

Example 1. To prove sin (A+B)=sin C.

A+B+C=180° .. A+B=180o – C,

and .. sin (A+B)=sin (180° - C)=sin C.

[p. 104.]

C

2'

Find A from each of the six following equations, A being an

Prove the following statements, A, B, C being the angles of a

(15) tan A-cot B= cos C. sec A. cosec B.

(25) sin 24+ sin 2B+ sin 2C=4 sin A. sin B. sin C.

Α

B

=

2

2

(26)

sin A. cos A - sin B. cos B+ sin C. cos C

=2 cos A. sin B. cos C.

(27) sin (B+C-A)-sin (C+A-B) + sin (A+B-C)

(28) cos 24+ cos 2B+cos 2C= -1-4 cos A. cos B. cos C.

(29)

sin2 A - sin2 B+ sin2 C-2 sin A. cos B. sin C. (30) cos (B+C-A)+cos (C+A-B) - cos (A+B-C)+1

(32) tan A+tan B+tan C=tan A. tan B. tan C.

A

(33) tan, tan+tan. tan+tan 4. tan-1.

2