Sidebilder
PDF
ePub

PROBLEM III.

Given a Leg and its opposite Angle, to find the other Angle, the other Leg, and the Hypothenuse.

Example.

Let the leg AC, of the spherical triangle ABC, be 56:30:40, and the angle B 70:23:35; required the angle A, the leg BC, and the hypothenuse AB?

To find the angle A:

[blocks in formation]

Here the three circular parts which enter the proportion, are the given angle B, the given leg A C, and the required angle A; and since the angle B is disjoined from the other two parts by the intervention of the hypothenuse AB, it is the middle part, and the other two are the extremes disjunct, according to rule 2, page 183; therefore, by equation 2, page 183,

Radius co-sine of the angle B sine of the angle Ax co-sine of the leg A C.

And since AC is connected with the required part, it is to be the first term in the proportion. Hence,

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

Note. The angle A is ambiguous, since it cannot be determined, from the parts given, whether it is acute or obtuse.

To find the leg BC:

The three circular parts concerned in this case, are the legs A C and B C, and the given angle A; and since the right angle never separates the legs, BC is the middle part, and AC and the angle B are the extremes conjunct, by rule 1, page 183; therefore, by equation 1, page 183,

Radius sine of the leg BC= tangent leg AC x co-tangent angle B. Now, since radius is connected with the required term, it is to stand first in the proportion. Hence,

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

To the leg BC = {142.34.33"} Log. sine =

9.731119

Note. The leg BC is ambiguous, since it cannot be determined, from the parts given, whether it is acute or obtuse.

To find the hypothenuse A B:

Here the given leg A C is the middle part, because it is disjoined from the other two circular parts concerned, by the intervention of the angle A: hence the angle B and the hypothenuse A B are extremes disjunct; therefore,

=

Radius sine of leg AC sine of hyp. AB × sine of angle B.

And since the angle B is connected with the required term, it is to stand first in the proportion. Hence,

[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]

Note. The hypothenuse AB is ambiguous; that is, it may be either acute or obtuse, from the parts given.

62:17:30 Log. sine =

9.947103

PROBLEM IV.

Given a Leg and its adjacent Angle, to find the other Angle, the other Leg, and the Hypothenuse.

Example.

Let the leg AC, of the spherical triangle ABC, be 68:29:45, and the angle A 74:45:15; tequired the angle B, the leg B C, and the hypothenuse AB?

[blocks in formation]

To find the Angle B:

Here the circular parts concerned are, the leg AC, the given angle A, and the required angle B; and since the angle B is disjoined from the other two parts by the hypothenuse A B, it is the middle part, and the other two are the extremes disjunct, by rule 2, page 183; therefore, by equation 2, page 183,

Radius co-sine angle B sine of angle Ax co-sine leg A C.

Now, since radius is connected with the required term, it is to stand first in the proportion. Hence,

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

Note.-The angle B is acute, or of the same affection with its opposite given leg A C.

To find the Leg BC:

In this case, since the right angle never separates the legs, the three circular parts are joined together: hence the leg AC is the middle part, and the leg BC and the angle A are the extremes conjunct, according to rule 1, page 183; therefore, by equation 1, page 183,

Radius sine of leg AC = co-tangent angle A × tangent of leg B C. And since the angle A is connected with the required part, it is to be the first term in the proportion. Hence,

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

Note. The leg B C is acute, or of the same affection with its opposite given angle A.

To find the Hypothenuse AB :

In this case, since the three circular parts which enter the proportion are joined together, the given angle A is the middle part, and the leg A C and the hypothenuse A B are the extremes conjunct: therefore,

Radius co-sine of angle A = tangent of leg AC × co-tangent hypothenuse A B.

Now, the leg A C, being connected with the required part, is therefore to be the first term in the proportion. Hence,

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

To the hypothenuse A C=84: 5: 6 Log. co-tangent = 9.015381

Note. The hypothenuse is acute, because the given leg and angle are of the same affection.

PROBLEM V.

Given the two Legs, to find the Angles and the Hypothenuse.

[merged small][ocr errors]

Let the leg AC, of the spherical triangle ABC, be 70:10:20%, and the leg B C 76:38:40"; required the angles A and B, and the hypothenuse AB?

To find the Angle A:

70: 10:20

[blocks in formation]

Here, since the right angle never separates the legs, the leg A C is the middle part, and the leg BC and the required angle A are the extremes conjunct, agreeably to rule 1, page 183; therefore, by equation 1, page 183, Radius x sine leg AC = tangent leg BC x co-tangent angle A. Now, since the leg B C is connected with the required part, it is to be the first term in the proportion. Hence,

[blocks in formation]

Note. The angle A is acute, or of the same affection with its opposite given leg B C.

To find the Angle B:

In this case the leg BC is the middle part, and the leg AC and the'

required angle B are the extremes conjunct, according to rule 1, page 183; therefore, by equation 1, page 183,

Radius x sine of the leg BC tangent of leg AC x co-tangent angle B.

And since the leg A C is connected with the required part, it is to be the first term in the proportion. Hence,

[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]

9.545083

To the angle B = 70:40 51" Log. co-tangent =

[ocr errors]

Note. The angle B is acute, or of the same affection with its opposite given leg A C.

To find the Hypothenuse A B:

Here the hypothenuse A B is the middle part, because it is disjoined from the legs by the angles A and B: hence AC and BC are extremes disjunct, agreeably to rule 2, page 183; therefore, by equation 2, page 183, Radius x co-sine hypothenuse AB co-sine leg AC x co-sine leg BC. And radius, being connected with the middle part, is therefore to be the first term in the proportion. Hence,

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

Note. The hypothenuse AB is acute, because the given legs A C and BC are of the same affection.

PROBLEM VI.

Given the two Angles, to find the Hypothenuse and the two Legs.

Example.

Let the angle A, of the spherical triangle ABC, be 50:10:20%, and the angle B 64:20:25"; required the legs A C and B C, and the hypothenuse AB?

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

.

« ForrigeFortsett »