Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[merged small][ocr errors][merged small]

sine (c+c)

And, Tang (45°+}a)=±√ sine (ce)

CASE XI. Given a side of a right-angled spherical triangle, and its opposite angle, to find the adjacent angle.

[merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

CASE XII. Given a side of a right-angled spherical triangle, and its opposite angle, to find the hypothenuse.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

Also, tang (45°+6)=± √tang (c+c). cot

Cosec c

cosec c.r

Cosec C

(C—c). And, tang (45°+6)=± √tang ≥ (▲+a). cot } (A—a).

CASE XIII. Given the two sides, or legs, of a right-angled spherical triangle, to find an angle.

sine c.r

sine c.

cot a

cot a. r

SOLUTION. Cot A

tang a

cosec c

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

CASE XIV. Given the two sides, or legs, of a right-angled spherical triangle, to find the hypothenuse.

tang c. cosec a

γ

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

COS A.7

sine c.r

COS A
COS C.r

sine A sine A.

CASE XV. Given two angles of a right-angled spherical triangle, to find a side, or leg.

SOLUTION. Cos a=

Sec a=

And, Cos c=

Sec

sine c

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]
[blocks in formation]

CASE XVI. Given two angles, of a right-angled spherical triangle, to find the hypothenuse.

sec c

sec c.r

[ocr errors]

C- A

— 45°

[ocr errors]
[blocks in formation]
[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

GENERAL OBSERVATIONS ON THE SPECIES AND AMBIGUITY OF THE CASES.

(A) The species of the sides and angles may be determined from the equations produced by Baron Napier's Rules, or from the preceding formulæ, by attending to the signs of the quantities which compose the equations or formulæ.

The sides which contain the right angle are each of the same species as their opposite angles, viz. a is of the same species with A, and b is of the same species with B. (R. 145.)

It may be proper to observe that where a quantity is to be determined by the sines only, and a side or angle opposite to the quantity sought does not enter into the equation, the case will be ambiguous, thus in the x11th case, where sine b = sine a. the hypothenuse is ambiguous.

sine A
Again, in the 11d case, where sine a

sine A. sine b

[ocr errors]

the sine of a is evidently determinate, because it is of the same species with a which is a given quantity.

(B) When an unknown quantity is to be determined by its cosine, tangent, or cotangent, the sign of this value will always determine its species; for, if its proper sign be+, the arc will be less than 90°; if the proper sign be-, the arc will be greater than 90°. (K. 100.)

(C) Again, in Case vith, where rad x cos b=cos cx cos a, it is obvious that the three sides are each less than 90°, or that two of them are greater than 90°, and the third less; as no other combination can render the sign of cos c x cos a like that of cos b as the equation requires.*

QUADRANTAL TRIANGLES.

(D) Any spherical triangle of which A, B, C, are the angles, and a, b, c, the opposite sides, may be changed into a spherical triangle of which the angles are supplements of the sides a, b, c,

* Legendre's Geometry, 6th Edition, page 381.

and the sides supplements of the angles A, B, C, (U 137.) viz. if we call A, B, C' the angles of the supplemental triangle, and a', b, c the sides opposite to these angles, we shall have

A180°-a; B'=180°-b; c=180°-c

a=180°-A; b′=180°—B; c'′=180°-c

Hence it is plain, that if a spherical triangle, has a side b equal to a quadrant, the corresponding angle B' of the supplemental triangle will be a right angle, and since there are always three given parts in a triangle, the supplemental triangle will be a right-angled triangle, having two parts given to find the rest; consequently, by finding the required parts in the supplemental right-angled triangle, the different parts of the quadrantal triangle will be known.

(E) Formulæ might have been inserted for solving the different cases of quadrantal triangles, but this would be making an increase of formulæ to very little purpose, since all quadrantal triangles are easily turned into right-angled triangles.

SOLUTIONS OF THE DIFFERENT CASES OF OBLIQUE-ANGLED SPHERICAL TRIANGLES.

CASE I. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the angle opposite to the other.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

CASE II. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the angle contained between these sides.

SOLUTION. Find the angle opposite to the other given side

by Case I.

[blocks in formation]
[blocks in formation]

CASE III. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the other side.

SOLUTION. Find the angle opposite to the other given side by Case I.

sine (C+B)

Then, Tanga =

sine (CB)

tang (c~b)

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

CASE IV. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the other opposite side.

ཝཱ

[blocks in formation]

CASE V. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the side adjacent to these angles.

« ΠροηγούμενηΣυνέχεια »