Spherical Trigonometry, for the Use of Colleges and Schools

On small variations in the parts of a Spherical Triangle

Excerpts

On small variations in the parts of a Spherical Triangle
It is sometimes important to know what amount of error will be introduced into one of the calculated parts of a triangle by reason of any small error which may exist in the given parts.
On small variations in the parts of a Spherical Triangle
*A side and the opposite angle of a spherical triangle remain constant: determine the connexion between the small variations of any other pair of elements*.
On small variations in the parts of a Spherical Triangle
Suppose $C$ and $c$ to remain constant.
On small variations in the parts of a Spherical Triangle
then we require the ratio of $\delta a$ to $\delta b$ when both are extremely small.
On small variations in the parts of a Spherical Triangle
If $A$ and $C$ are constant, and $b$ be increased by a small quantity, shew that $a$ will be increased or diminished according as $c$ is less or greater than a quadrant.

Equations

On small variations in the parts of a Spherical Triangle
\cos c = \cos a \,\cos b + \sin a \,\sin b \,\cos C
The cosine of side c equals the cosine of a times the cosine of b plus the sine of a times the sine of b times the cosine of the angle C, for a spherical triangle.
On small variations in the parts of a Spherical Triangle
\delta a \,\cos B + \delta b \,\cos A = 0
The small variations of the sides a and b, when C and c are constant, satisfy this relation with the cosines of the opposite angles B and A.
On small variations in the parts of a Spherical Triangle
\delta A \,\cos b + \delta B \,\cos a = 0
By the polar triangle, the small variations of the angles A and B are connected by the cosines of the sides b and a.
On small variations in the parts of a Spherical Triangle
\cos (a + \delta a) = \cos a - \sin a \,\delta a
To first order in the small increment, the cosine of a side increased by delta a is the cosine minus sine times delta a.
On small variations in the parts of a Spherical Triangle
\sin (a + \delta a) = \sin a + \cos a \,\delta a
To first order in the small increment, the sine of a side increased by delta a is the sine plus cosine times delta a.
On small variations in the parts of a Spherical Triangle
\sin A \sin c = \sin C \,\sin a
The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.
On small variations in the parts of a Spherical Triangle
\delta A \cot A = \delta a \cot a
For constant side c and opposite angle C, the small variations of the side a and its opposite angle A are related through their cotangents.
On small variations in the parts of a Spherical Triangle
\cot C \sin B = \cot c \sin a - \cos B \cos a
A four-part relation among the sides c, a and the angles C, B of a spherical triangle, involving cotangents, sines and cosines.
On small variations in the parts of a Spherical Triangle
\delta B \cos A = - \delta a \cot b \sin B
With C and c constant, the small variation of the angle B is tied to that of the side a by cosine of A and cotangent of b.
On small variations in the parts of a Spherical Triangle
-\frac{\cos A}{\sin C} \delta B = \frac{\cos b}{\sin c} \delta a
The intermediate relation between the small variations of the angle B and the side a, with the sines and cosines of the triangle's parts, from which the result is obtained.
On small variations in the parts of a Spherical Triangle
\frac{\delta a}{\surd{(1 - n^2 \sin^2 a)}} + \frac{\delta b}{\surd{(1 - n^2 \sin^2 b)}} = 0
Example 1: if C and c are constant, the small increments of a and b, scaled by the square roots involving n, sum to zero.
On small variations in the parts of a Spherical Triangle
n = \frac{\sin C}{\sin c}\,
Definition of n as the ratio of sin C to sin c, used in Example 1.
On small variations in the parts of a Spherical Triangle
\sin C \delta b = \sin a \delta B
Example 3: with A and c constant, the small variations of b and B are related through sin C and sin a.
On small variations in the parts of a Spherical Triangle
\delta b \sin C = -\delta C \tan a
Example 3: with A and c constant, the small variations of b and C are related through sin C and tan a.
On small variations in the parts of a Spherical Triangle
\delta a \tan C = \delta B \sin a
Example 3: with A and c constant, the small variations of a and B are related through tan C and sin a.
On small variations in the parts of a Spherical Triangle
\delta a \tan C = -\delta C \tan a
Example 3: with A and c constant, the small variations of a and C are related through tan C and tan a.
On small variations in the parts of a Spherical Triangle
\delta b \cos C = \delta a
Example 3: with A and c constant, the small variations of b and a are related through cos C.
On small variations in the parts of a Spherical Triangle
\delta B \cos a = -\delta C
Example 3: with A and c constant, the small variations of B and C are related through cos a.
On small variations in the parts of a Spherical Triangle
\delta B \tan C = \delta C \tan B
Example 4: with b and c constant, the small variations of B and C are related through their tangents.
On small variations in the parts of a Spherical Triangle
\delta a \cot C = -\delta B \sin a
Example 4: with b and c constant, the small variations of a and B are related through cot C and sin a.
On small variations in the parts of a Spherical Triangle
\delta a = \delta A \sin c \sin B
Example 4: with b and c constant, the small variation of a equals that of A times sin c sin B.
On small variations in the parts of a Spherical Triangle
\delta A \sin B \cos C = -\delta B \sin A
Example 4: with b and c constant, the small variations of A and B are related through sines and the cosine of C.
On small variations in the parts of a Spherical Triangle
\delta b \tan c = \delta c \tan b
Example 5: with B and C constant, the small variations of b and c are related through their tangents.
On small variations in the parts of a Spherical Triangle
\delta A \cot c = \delta b \sin A
Example 5: with B and C constant, the small variations of A and b are related through cot c and sin A.
On small variations in the parts of a Spherical Triangle
\delta A = \delta a \sin b \sin C
Example 5: with B and C constant, the small variation of A equals that of a times sin b sin C.
On small variations in the parts of a Spherical Triangle
\delta a \sin B \cos c = \delta b \sin A
Example 5: with B and C constant, the small variations of a and b are related through sines and the cosine of c.

Problems

Exercise XI

Exercise XI, problem 1, p. 105
In a spherical triangle, if $C$ and $c$ remain constant while $a$ and $b$ receive the small increments $\delta a$ and $\delta b$ respectively, shew that a(1 - n^2 ^2 a) + b(1 - n^2 ^2 b) = 0 where n = Cc .
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Exercise XI, problem 2, p. 105
If $C$ and $c$ remain constant, and a small change be made in $a$, find the consequent changes in the other parts of the triangle. Find also the change in the area.
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Exercise XI, problem 3, p. 105
Supposing $A$ and $c$ to remain constant, prove the following equations, connecting the small variations of pairs of the other elements: gather* C b = a B, b C = -C a, a C = B a, a C = -C a, b C = a, B a = -C. gather*
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Exercise XI, problem 4, p. 105
Supposing $b$ and $c$ to remain constant, prove the following equations connecting the small variations of pairs of the other elements: align* & B C = C B, & & a C = -B a, & a = A c B, & & A B C = -B A. align*
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Exercise XI, problem 5, p. 105
Supposing $B$ and $C$ to remain constant, prove the following equations connecting the small variations of pairs of the other elements: align* b c &= c b, & A c &= b A, A &= a b C, & a B c &= b A. align*
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Exercise XI, problem 6, p. 105
If $A$ and $C$ are constant, and $b$ be increased by a small quantity, shew that $a$ will be increased or diminished according as $c$ is less or greater than a quadrant.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles