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bers 1, 2, 4. Consequently there are given the sum of the three angles 180° (32. 1), and their ratio, to find the angles.
Let x=4C, then 2 x B, and 4 x ≤ A.
Hence 7 x = 180°, therefore x == therefore
Now the sides of a plane triangle are respectively as the sines of their opposite angles. Hence there are given the sum of three numbers 100, and their ratio, to find the numbers.
Let a= trees, then
and y = least side, or distance between two
yxs. 2 a
y X s. 4 a
s. a: s. 2 a :: y:
and s. a: s. 4 ay
= longest side.
therefore y X s. a+yx s. 2a+yx s. 4a=100 X s. a,
This formula is not adapted to calculation by logarithms; therefore natural sines must be applied, as follows.
=s. 102° 51′26′′s.sup. 77°8′ 34′′ =•97493
18. Let A and B be two inaccessible objects, whose distance is required. At two stations C and D, from which the objects and also the other station can be seen, the angles ACB 62° 12', BCD 41° 8', and ADB 60° 49', ADC 34° 51', are taken
with an instrument; and the distance CD between the stations is found to be 562 yards. Answer. 729-7 yards.
Note. This problem is useful in taking surveys of coasts or harbours.
19. The angle of elevation of a certain tower was observed to be 20°. The observer intended to measure his distance from the bottom of the tower, to determine its height; but when he had measured 85 feet in a direct line toward the tower, he was stopped by a ditch; therefore he again took the altitude of the tower, and found it to be 31° 34'. Required the height of the tower, and the distance of the observer from the bottom, when he was stopped by the ditch.
20. Wanting to know the
height of a hill OC, and its
and ABO 79° 29′; also at A I took the elevation of the hill OAC 4° 51'. Required the distance AC, and the height CO. Answer. AC 344-6 yards, and CO 29.2.
21. There is a certain mountain, the height of which above the neighbouring plain is required. To determine its height two stations were taken upon the plain in sight of the highest cliff, and in sight of each other. At the first station A the angle of elevation of the cliff was observed to be 22° 49′, and the angle between the cliff and the second station B was found to be 39° 10'. At the second station B the angle between the cliff and the first station A was observed to be 48° 45'. Lastly, the distance between the two stations was found to be 240 yards. Required the height of the cliff, and its distance from the nearer station; also that point in the base line (produced if necessary) which is nearest to the cliff.
It is evident that the station B is nearer the cliff than A, because the angle at B is greater than the angle at A, and therefore the side opposite to B is greater than the side opposite to A. Moreover the point in the base produced, where the angle between the cliff and the first station is a right angle, will be nearest to the cliff.
22. If A and C be two stations on sloping ground, 410 yards distant, O an object on the top of a hill, the angle OCA
79° 29', the angle OAC = 63° 11', the angle of elevation at A6° 36', at C=5° 22'; what are the horizontal distances of the object from the two stations, and its height above the level of each station?
= 56.43 yards, and CB 600.73 yards.
23. At a mile-stone N, on the ascending road NS, a person observed the angle SNW between the next mile-stone S and the windmill W, on the top of a hill, and found it to be 46° 37′; he also took the angle of elevation of W 3° 49′. At the next mile-stone S he took the angle NSW 91° 4′ between the first mile-stone and the mill. Required the horizontal distance NP, and the height PW of the object. Answer. NP2608 yards, and PW=174.
24. To complete some chorographical observations about Cambridge, in England, it was necessary to measure exactly the distance between the steeple of St. Mary's church and the observatory at Trinity college; but that distance could not be measured in a direct line by reason of the houses between these two objects. Therefore two stations were chosen in the fields behind Trinity college, from each of which the observatory, St. Mary's, and the other station could be all seen at once. At the first station the angle made by the top of the observatory and the top of St. Mary's steeple was 14° 34', the angle between the top of the observatory and the second station was 60° 50′, and the angle between the top of St. Mary's steeple and the second station was 46° 16'. At the second station the angle between the top of the observatory and the first station was 96° 44′, the angle between the top of St. Mary's and the first station was 115° 23'. Lastly, the base line or distance between the two stations was exactly measured, and found to be 908.36 feet.
What is the horizontal distance between the observatory and St. Mary's steeple from these data?
1. Answer. If the two buildings were of the same height the distance would be 674.62 feet.
2. If the observatory be 73.50 feet high, and St. Mary's steeple 97.25 feet high, the distance is 674.20 feet, which differs from the other distance by less than half a foot.
This problem is similar to prob. 12, page 47, and may be resolved in the same manner.
I. Of the Signs of Trigonometrical Lines.
97. The several changes in the algebraic signs of all the lines described in and about a circle must be particularly observed in the application of trigonometry to the solution of astronomical and physical problems. We shall therefore trace all the changes of the signs of those lines particularly, as follows. See plate, figure 1.
Let AHDL be a circle having the sines, tangents, &c. represented as in the figure. Suppose one extremity A of an arc AB to remain fixed, while the other extremity B passes successively over the circumference of the circle, from A through the points H, D, L, to A again.
The sine begins at A, and increases from 0 (nothing) during the first quadrant AH, till it becomes equal to radius at the point H. Then it decreases during the second quadrant HD, till it again becomes O at the end of it. After this the sine passes to the other side of the diameter AD; and therefore, being reckoned affirmative before, is now to be considered negative. During the third quadrant DL the sine Bf increases till it becomes equal to radius, but is negative, or equal to radius. During the fourth quadrant LA it decreases from radius till it becomes 0, when the arc is 360 degrees. After this the sine is affirmative, and changes continually as at first, in every revolution of the point B round the circumference of the circle.
It appears from an inspection of the figure, that the sine BF increases faster in the first part AB of the quadrant AH, than when it approaches near the end of it, at H; and, on the contrary, that it decreases slower in the first part Hb of the second quadrant HD, than when it arrives near the end of it, at
* The learner may omit this section at the first reading of plane trigonometry, but must afterward read the whole with attention, if he intends to prosecute the study of mathematics, because it is necessary to be known in many parts of mechanical philosophy.
D. This variation of the rate of the increase or decrease of the sine arises from the convexity of the circle in certain positions of the sine.
98. The cosine is equal to radius, when the arc is 0, and continually decreases during the first quadrant AH, till it becomes O at the end of it, at H. The cosine being computed from the centre of the circle, will be negative after it passes the centre; therefore the cosine Cf, which lies in an opposite direction to the cosine CF, will be negative. This negative cosine increases during the second quadrant HD, at the end of which it is equal to radius. It decreases negatively during the third quadrant DL, at the end of which it is 0. In the fourth quadrant LA it becomes positive, and increases till it is equal to radius, as before.
Since the cosine of the arc AB is equal to the sine of its complement BH, it follows, reversedly, from what has been said respecting the variation of the rate of increase of the sines, that the cosine CF decreases slower in the first part AB of the quadrant AH, than when it approaches near the end of it, at H; and, on the contrary, that it increases faster in the first part Hb of the second quadrant HD, than when it arrives near the end of it, at D.
99. The tangent AT becomes negative as often as it meets the radius CB produced on the side of the point A or diameter AD opposite to that in which it is first drawn; and as this happens both when the arc AB becomes greater than AH, and also when, by its further increase, it is greater than the arc AHDL, it follows that the tangents of all arcs in the first and third quadrants AH, DL, are positive, and that the tangents of all arcs in the second and fourth quadrants HD, LA, are negative.
The tangent at the beginning of the arc is 0, and increases to infinity during the first quadrant AH; that is, no line can be assigned, how great soever its length, but an angle under 90 degrees may be found, whose tangent shall exceed that line; consequently the tangent has no limit to its increase, as the sine has. In the second quadrant HD the tangent At is negative, for the tangents being computed from the point A, the tangent At will be in a direction opposite to AT. During this quadrant the tangent decreases from an infinite negative to O. In the third quadrant DL it is again affirmative, and increases from 0 to infinity, as in the first quadrant. In the fourth quadrant LA it decreases from an infinite negative to O, as in the second quadrant; after which it becomes affirmative, and then increases as in the first quadrant.