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But in any fraction whose numerator does not diminish, the smaller the denominator, the greater the value of that fraction; and if the denominator continually diminishes, the value of the fraction continually increases.
Hence, tan XOP can be made larger than any assigned number by making the angle XOP approach near enough to 90°. This is what we mean when we say that
tan 90° is infinity, or, tan 90° = ∞.
41. The following table exhibits the above results:
The student may notice that the sine increases with the angle, while the cosine diminishes as the angle increases.
Also that the squares of the sines of 0°, 30°, 45°, 60°, and 90° are respectively 0, 1, 4, 1, and, and that the squares of the cosines of the same angles are, ,, and 0.
If A = 90°, B = 60°, C = 30°, D= 45°, prove the following:
1. cos2 B sin2 B = 1 − 2 sin2 B.
cos D + sin B. sin D)2 = 1 + cos C.
cos Csin C cos D)2 = 1
6. tan 60° 1.732 ....
9. tan 30°.5773....
7. sin 45°.7071 .........
42. The actual measurement of the line joining two points which are any considerable distance apart, is a very tedious and difficult operation, especially when great accuracy is required; while the accurate measurement of an angle can, with proper instruments, be made with comparative ease and quickness.
43. A Sextant is an instrument for measuring the angle between the two lines drawn from the observer's eye to each of two distant objects respectively.
A Theodolite is an instrument for measuring angles in a horizontal plane; also for measuring "angles of elevation" and "angles of depression."
44. The angle made with the horizontal plane, by the line joining the observer's eye with a distant object, is called
(i.) its angle of elevation, when the object is above the observer; (ii.) its angle of depression, when the object is below the observer.1
45. Trigonometry enables us, by measuring certain angles, to deduce other distances from one known distance, or, by the measurement of a convenient line, to deduce by the measurement of angles the lengths of lines whose actual measurement is difficult or impossible.
46. For this purpose we require the numerical values of the Trigonometrical Ratios of the angles observed. Accordingly, mathematical tables have been compiled, giving these ratios. These tables constitute a sort of numerical Dictionary, in which we can find the
1 In measuring the angle of depression the telescope is turned from a horizontal position downwards. See Ex. VII. 3.
numerical values of the trigonometrical ratios of any required angle.
EXAMPLE 1. At a point 100 feet from the foot of a tower, the angle of elevation of the top of the tower is observed to be 60°. Find the height of the top of the tower above the point of observation.
Let O be the point of observation; let P be the top of the tower; let a horizontal line through O meet the foot of the tower at the point M. Then OM = 100 feet, and the angle MOP 60°. Let MP contain x feet.
Therefore the required height is 173.2.
... x = 100. √3 = 100 × 1.7320,
EXAMPLE 2. At a point 100 yards from the foot of a building, I measure the angle of elevation of the top, and find that it is 23° 15'; what is the height of the building?
As in Example 1, let the height be x yards.
tan 23° 15'.
From the table of tangents we find that tan 23° 15′ = .4296339.
Hence x 100 x .4296339 = 42.96339.
The height of the building = 43 yards, nearly. Ans.
EXAMPLE 3. A flagstaff, 25 feet high, stands on the top of a cliff; from a point on the seashore the angles of elevation of the highest and lowest points of the flagstaff are observed to be 47° 12′ and 45° 13′ respectively; find the height of the cliff.
Let O be the point of observation, PQ the flagstaff.
Let a horizontal line through O meet the vertical line PQ produced in M. QP 25 feet, MOP = 47° 12', MOQ = 45° 13'.
x + 25
.. x =
Therefore the cliff is 348 feet high.
tan 47° 12' 1.0799018, and tan 45° 13′ = 1.0075918.
= tan 47° 12',
tan 45° 13'.
tan 47° 12'
tan 45° 13'
= 348, nearly.
NOTE. The answers are given correct to three significant figures.
1. At a point 179 feet in a horizontal line from the foot of a column, the angle of elevation of the top of the column is observed to be 45; what is the height of the column?
2. At a point 200 feet from, and on a level with the base of a tower, the angle of elevation of the top of the tower is observed to be 60°; what is the height of the tower?
3. From the top of a vertical cliff, the angle of depression of a point on the shore 150 feet from the base of the cliff, is observed to be 30; find the height of the cliff.
4. From the top of a tower 117 feet high the angle of depression of the top of a house 37 feet high is observed to be 30-; how far is the top of the house from the tower?
5. A man 6 feet high stands at a distance of 4 ft. 9 in. from a lamp-post, and it is observed that his shadow is 19 feet long; find the height of the lamp. 6. The shadow of a tower in the sunlight is observed to be 100 feet long, and at the same time the shadow of a lamp-post 9 feet high is observed to be 3√3 feet long. Find the angle of elevation of the sun, and the height of the
7. From a point P on the bank of a river, just opposite a post on the other bank, a man walks at right angles to PQ to a point R so that PR is 100 yards; he then observes the angle PRQ to be 32° 17′; find the breadth of the river. (tan 32° 17′ = .6317667.)
8. A fine wire 300 feet long is attached to the top of a spire and the inclination of the wire to the horizon when held tight is observed to be 40°; find the height of the spire. (sin 40° = .6428.)
9. A flagstaff 25 feet high stands on the top of a house; from a point on the plane on which the house stands the angles of elevation of the top and bottom of the flagstaff are observed to be 60° and 45° respectively; find the height of the house above the point of observation. 34
10. From the top of a cliff 100 feet high, the angles of depression of two ships at sea are observed to be 45° and 30° respectively; if the line joining the ships points directly to the foot of the cliff, find the distance between the ships.
11. A tower 100 feet high stands on the top of a cliff; from a point on the sand at the foot of the cliff the angles of elevation of the top and bottom of the tower are observed to be 75° and 60° respectively; find the height of the cliff. (tan 75° = 2 + √3).
12. A man walking along a straight road observes at one milestone a house in a direction making an angle 30° with the road, and that at the next milestone the angle is 60°; how far is the house from the road?
13. A man stands at a point A on the bank AB of a straight river and observes that the line joining A to a post C on the opposite bank makes with AB an angle of 30°. He then goes 400 yards along the bank to B and finds that BC makes with BA an angle of 60°; find the breadth of the river.
14. From the top of a hill the angles of depression of the top and bottom of a flagstaff 25 feet high at the foot of the hill are observed to be 45° 13' and 47° 12' respectively; find the height of the hill. (tan 45° 13′ = 1.0075918. (tan 47° 12' 1.0799018.)
15. An isosceles triangle of wood is placed on the ground in a vertical position facing the sun. If 2 a be the base of the triangle, b its height, and 30° the altitude of the sun, find the tangent of half the angle at the apex of the shadow.
16. The length of the shadow of a vertical stick is to the length of the stick as 3:1. If the stick be turned about its lower extremity in a vertical plane, so that the shadow is always in the same direction, find what will be the angle of its inclination to the horizon when the length of the shadow is the same as before.
17. What distance in space is travelled in an hour in consequence of the earth's rotation, by a person situated in latitude 60°? (Earth's radius = 4000 miles.)