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sines may be computed to any arc; and the tangents and other trigonometrical lines, by means of the expressions in art. 4, &c. Er. 3. Find the sum of all the natural sines to every mi. nute in the quadrant, radius = 1. In this problem the actual addition of all the terms would be a most tiresome labour: but the solution by means of equation xxvii, is rendered very easy. Applying that theorem to the present case, we have sin (A + 3nb) = sin 45°, sing (n+1)b =sin 45°0'30", and sin b-sin 30°. Therefore nor = 3438:2467465 the sum required. From another method, the investigation of which is omit. ted here, it appears that the same sum is equal to 3 (cot 30° + 1). Er. 4. Explain the method of finding the logarithmic sines, cosines, tangents, secants, &c. the natural sines, cosines, &c. being known. The natural sines and cosines being computed to the radius unity, are all proper fractions, or quantities less than unity, so that their logarithms would be negative. To avoid this, the tables of logarithmic sines, cosines, &c. are computed to a radius of 10000000000, or 10"; in which case the logarithm of the radius is 10 times the log of 10, that is, it is 10. Hence, if s represent any sine to radius 1, then 10"X s = sine of the same arc or angle to rad 10". And this, in logs is, log 10"s- 10 log 10+log s−10+log s. The log cosines are found by the same process, since the cosines are the sines of the complements. The logarithmic cypressions for the tangents, &c. are de. duced thus : Tan = rad *. Theref, log tan =log rad+log sin–log

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Therefore, log vers sin = log2+2 log sin 4 arc-10. Er. 5. Given the sum of the natural tangents of the angles A and B of a plane triangle-3-100.1988, the sum of the tangents of the angles B and c=38765577, and the continued product, tan A. tan B. tan c=53047057 : to find the angles A, B, and c.

It has been demonstrated in art. 36, that when radius is unity, the sum of the natural tangents of the three angles of a plane triangle is equal to their continued product. Hence the process is this :

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Remains tan A . . . . . . == 14281.480 = tan 55°.
Consequently, the three angles are 55°, 60°, and 65°.

Er. 6. There is a plane triangle, whose sides are three consecutive terms in the natural series of integer numbers, and whose largest angle is just double the smallest. Requir. ed the sides and angles of that triangle 1

If A, B, c, be three angles of a plane triangle, a, b, c, the sides respectively opposite to A, B, c; and s = a + b + c, Then from equa. III and xxxiv, we have

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A geometrical construction of this example is given at p. 59, Gregory's Trigonometry. Ex. 7. Demonstrate that sin 18° - cos 72° is - + R (— 1 + v 5), and sin 54° = cos 36° is - R (1 + v5). Er. 8. Demonstrate that the sum of the sines of two arcs which together make 60°, is equal to the sine of an arc which is greater than 60° by either of the two arcs: Ex. gr. sin 3+sin 59°57' = sin 60°3'; and thus that the tables may be continued by addition only. Ex. 9. Show the truth of the following proportion: As the sine of half the difference of two arcs, which together make 60°, or 90°, respectively, is to the difference of their sines; so is 1 to V3, or V2, respectively. Er. 10. Demonstrate that the sum of the squares of the sine and versed sine of an arc, is equal to the square of double the sine of half the arc. Ez. 11. Demonstrate that the sine of an arc is a mean proportional between half the radius and the versed sine of double the arc. Er. 12. Show that the secant of an arc is equal to the sum of its tangent and the tangent of half its complement. Er. 13. Prove that, in any plane triangle, the base is to the difference of the other two sides, as the sine of half the sum of the angles at the base, to tho sine of half their difference : also, that the base is to the sum of the other two sides, as the cosine of half the sum of the angles at the base, to the cosine of half their difference. Er. 14. How must three trees A, B, C, be planted, so that the angle at A may be double the angle at B, the angle at B double that at c ; and so that a line of 400 yards may just go round them 1 Er. 15. In a certain triangle, the sines of the three angles are as the numbers 17, 15, and 8, and the perimeter is 160. What are the sides and angles? Er. 16. The logarithms of two sides of a triangle are 2.2407293 and 2.5378.191, and the included angle is 37°20'. It is required to determine the other angles, without first finding any of the sides! Ez. 17. The sides of a triangle are to each other as the fractions, 1, 4, #: what are the angles 1 Er. 18. Show that the secant of 60°, is double the tan. gent of 45°, and that the secant of 45° is a mean proportional between the tangent of 45° and the secant of 60°.

Ez. 19. Demonstrate that 4 times the rectangle of the

sines of two arcs, is equal to the difference of the squares of the chords of the sum and difference of those arcs. Ec. 20. Convert the equations marked xxxiv into their equivalent logarithmic expressions; and by means of them and equa. Iv, find the angles of a triangle whose sides are 5, 6, and 7. Er. 21. Find the arc whose tangent and cotangent shall together be equal to 4 times the radius. Er. 22. Find the arc whose sine added to its cosine shall be equal to a ; and show the limits of possibility. Er. 23. Find the arc whose secant and cotangent shall be equal. Er. 24. If one angle A of a right-angled plane triangle A, B, c, be given, (B being be the right angle), and the area or surface be given = s. Demonstrate that AB = V (2s cot A) . . . b c = V (2s tan A), and Ac = 2 v (s sec 2A) Er. 25. Demonstrate, 1

1. That sin A = vào) - V (#–; cos 2A) f 2 l

== *A-Hitan R= cot A+-tan #A

= 2 sin” (45°-F A) — 1 = 1 –2 sin” (45°–4A).

t 1—cos?A 2. That tan A = v'(z=-1) = v'i Fo V(1—cos^a) sin A = −77- - ZT-iño, _ 1-cos 2A sin 2A ~ Tin GT = I-Ecoso. 2 cot . A 2 (an A

F coola-i T 1–tano

===His = cot a -2 cot 2A

== #[tan (45°-H #A) — tan (45° – #A)].

Wol. II. 5


( , , | 4. SECTION I.

General Properties of Spherical Triangles.

ART. 1. Def. 1. Any portion of a spherical surface bounded by three arcs of great circles, is called a Spherical Triangle.

Def. 2. Spherical Trigonometry is the art of computing the measures of the sides and angles of spherical triangles.

Def. 3. A right-angled spherical triangle has one right angle : the sides about the right angle are called legs ; the side opposite to the right angle is called the hypothenuse.

Def. 4. A quadrantal spherical triangle has one side equal to 90° or a quarter of a great circle.

Des. 5. Two arcs or angles, when compared together, are said to be alike, or of the same affection, when both are less than 90°, or both are greater than 90°. But when one is greater and the other less than 90°, they are said to be unlike, or of different affections.

ART. 2. The small circles of the sphere do not fall under consideration in Spherical Trigonometry; but such only as have the same centre with the sphere itself. And hence it is that spherical trigonometry is of so much use in Practical Astronomy, the apparent heavens assuming the shape of a concave sphere, whose centre is the same as the centre of the earth.

3. Every spherical triangle has three sides and three angles: and if any three of these six parts be given, the remaining three may be found, by some of the rules which will be investigated in this chapter.

4. In plane trigonometry, the knowledge of the three angles is not sufficient for ascertaining the sides: for in that case the relations only of the three sides can be obtained, and not their absolute values: whereas, in spherical trigonometry, where the sides are circular arcs, whose values depend on their proportion to the whole circle, that is, on the number of degrees they contain, the sides may always be determined when the three angles are known. Other remarkable dif. ferences between plane and spherical triangles are, 1st. That in the former, two angles always determine the third ; while in the latter they never do. 2dly. The surface of a plane

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