28

CHAPTER V.

PROBLEMS ON SOLUTION OF RIGHT-ANGLED

TRIANGLES.

31. We may now turn the preceding chapters to practical account in solving easy exercises concerning heights and distances.

The only new process is the construction of diagrams to illustrate each problem. If these be made as clear and as accurate as is consistent with expedition, the care bestowed on them will be amply compensated when they become difficult in any degree.

32. The student may require explanations of certain terms which will now come into frequent use.

The angular altitude, or angle of elevation, of an object above the eye of an observer, is the angle contained by two straight lines drawn from the eye, one of which is horizontal, and the other meets the object. The words, altitude and elevation, simply, are most generally used in Trigonometry to denote this angle.

The angle of depression, or depression simply, of an object below the eye of an observer, is the angle contained by two straight lines drawn from the eye, one of which is horizontal, and the other meets the object.

33. Mariner's Compass.-It is useful to know accurately what are called the Points of the Compass. They will be learnt best from the inspection of a "compass card;" a representation of which is here

34. After those of the last chapter, but few examples worked out will be required here.

(1.) At a distance of 36 yards a tower subtends an angle of 60°; and the observer's eye is 6 feet above the ground. What is the height of the tower?

Making BH the tower, and AC the observer, we have given, AB 36 yards,

=

AC 6 feet required BH.

=

2 yards, and

Draw CD parallel to AB.

In triangle DCH, because

=

√DCH = 60°;

АН

2 yds. +36/3 yds. ;

.. required height 64.352 yds.

=

B

(2.) A ship sailing due N. observes at noon two lighthouses in a line exactly E. At 20 minutes to 1 o'clock they are respectively S.E. and E. by S. At what rate does the ship sail, the lighthouses being 24 miles apart?

Given tan 11° 15′:

=

.2 nearly.

required Ss (= x miles suppose).

E

(3.) In ABC, having C a right angle, A and a+b-c are given; solve the triangle.

and

or

Let a+b-c = m;

.. a = c sin A, and b = c cos A;

.. m = c sin A+ c cos A− c = c (sin A+ cos A-1);

.. C =

C =

a =

1. How far must a person recede from a pillar 45 feet high, in order that it may subtend at his foot an angle of 60°?

2. The altitude of a tower, observed at the end of a horizontal base of 100 yards measured from its foot, is 30°; find its height.

3. A lighthouse was observed by a ship at sea to bear S.E.; after the ship had sailed N.E. for 12 miles, the lighthouse was observed to bear 15° E. of S. Find the distance of the lighthouse from each position of the ship.

4. The angle of elevation of the top of a steeple is 60° from a point on the ground. That of the top of the tower on which the steeple rests is 45° from the same point. What proportion does the height of the steeple bear to that of the tower?

5. The elevation of the top of a steeple is 45° from a point in the same level as its base, and is 30° from a point 30 feet directly above the former point; find the height and distance of the steeple.

6. A pole is fixed on the top of a mound, and the angles of elevation of the bottom and top of the pole are 30° and 60° re

spectively; prove that the height of the pole is twice the height of the mound.

7. Two straight roads, which cross one another, meet a canal at angles of 60° and 30° respectively. If it be 3 miles by the longer of the two roads, from the crossing to the canal; how far is it by the shorter? If there be a footpath which goes the shortest way to the canal, what is the distance by it?

8. A ladder 30 feet long reaches a window 24 feet high on one side of a street; turning over the top of the ladder to the other side of the street without moving its foot, it exactly reaches a window when its new position is at right angles to its former one. Required the width of the street and the height of the latter window.

9. A cruiser from a point N.E. of a harbour observes a blockaderunner in a direction 15° E. of S., running straight for the harbour with a N.W. course; which it pursues, and overtakes at the mouth of the harbour. Compare the rates of sailing.

10. From a ship sailing due S.E. at the rate of 7 miles per hour, a lighthouse is observed to bear N. 30° E., and after two hours its bearing is due N.; find the distance of the ship from the lighthouse at each observation.

11. A privateer descries a barque 12 miles off to the N.E. sailing with a N.W. course, and pursues and overtakes it running due N. for 8 miles, and W. 15° from N. for the remainder of the How far did each vessel run in the pursuit ?

race.

12. From the top of a ship's mast, 125 feet above its hull, the angle of depression of the hull of another ship is found to be 6° 50' 34". Find the distance between the ships.

13. A kite has attained an altitude of 36°, and 200 yards of string are out; required the perpendicular height of the kite, supposing the string to be perfectly stretched.

Given sin 36° = ⚫5878.

14. A castle wall, 20 feet high, is surrounded by a ditch; from the edge of which the altitude of the top of the wall is 60°. Find the width of the ditch, and also how far an observer must walk directly backward in order that the altitude of the wall may be 30°.

Given 31.732.

15. Starting from a point at which I perceive a steeple due E., and a tower due W., I travel for 5 miles and then observe the steeple to be S.E., and the tower to be 15° from S. How far is it from tower to steeple ?

16. From the top of a hill the angles of depression of two con

and so for the remaining ratios.

14. Relations between the Functions of Angles.-There are certain relations of great importance existing between these ratios; and the five results of the demonstrations which follow should be carefully noted, and committed to memory as formulæ.

In the right-angled triangle ABC,

.. cot A =

(5).