Spherical Trigonometry, for the Use of Colleges and Schools
On certain approximate Formul\ae
Excerpts
On certain approximate Formul\ae
If the sides of the triangle are small compared with the radius of the sphere, $EF$ will not differ much from $A$;
On certain approximate Formul\ae
This gives the *circular measure* of $\theta$
On certain approximate Formul\ae
The importance of Legendre’s Theorem in the application of Spherical Trigonometry to the measurement of the Earth’s surface has given rise to various developments of it which enable us to test the degree of exactness of the approximation.
On certain approximate Formul\ae
*If the sides of a spherical triangle be small compared with the radius of the sphere, then each angle of the spherical triangle exceeds by one third of the spherical excess the corresponding angle of the plane triangle, the sides of which are of the same length as the arcs of the spherical triangle.*
On certain approximate Formul\ae
It will be seen that in the above approximation the area of the spherical triangle is considered equal to the area of the plane triangle which can be formed with sides of the same length.
On certain approximate Formul\ae
Thus when the sides of the spherical triangle and the radius of the sphere are known, we can calculate the angles and sides of the chordal triangle.
On certain approximate Formul\ae
therefore $\dfrac{S}{r^2}$ is approximately equal to the spherical excess of the spherical triangle, and thus the theorem is established.
Equations
On certain approximate Formul\ae
\cos EF = \sin \frac{1}{2} b \sin \frac{1}{2} c + \cos \frac{1}{2} b \cos \frac{1}{2} c \cos AThe cosine of the chord-angle side EF equals a combination of the half-sides b/2 and c/2 and the angle A between the two sides of the triangle.
On certain approximate Formul\ae
\cos EF=\cos DE \cos DF + \sin DE \sin DF \cos AThe spherical cosine rule applied to the spherical triangle DEF, with angle A at D.
On certain approximate Formul\ae
\theta = \tan\tfrac12A \sin^2\tfrac14(b + c) - \cot\tfrac12A \sin^2\tfrac14(b - c)The small angle theta by which EF differs from A is approximately tan(A/2) sin^2((b+c)/4) minus cot(A/2) sin^2((b-c)/4); this gives its circular measure.
On certain approximate Formul\ae
2r\sin\dfrac{\alpha}{2r}The length of a side of the chordal triangle, from the arc alpha of the sphere of radius r.
On certain approximate Formul\ae
\cos A = \frac{\cos a - \cos b\cos c}{\sin b \sin c}The spherical cosine rule giving the cosine of angle A from the three sides a, b, c.
On certain approximate Formul\ae
1 - \frac{\alpha^2}{2r^2} + \frac{\alpha^4}{24r^4} - \ldotsSeries expansion of cos a in powers of alpha/r, where alpha is the arc corresponding to side a.
On certain approximate Formul\ae
\frac{\alpha}{r} - \frac{\alpha^3}{6r^3} +\ldotsSeries expansion of sin a in powers of alpha/r, where alpha is the arc corresponding to side a.
On certain approximate Formul\ae
\theta = \frac{\beta \gamma \sin A'}{6r^2}=\frac{S}{3r^2}Legendre's correction: the excess of the spherical angle A over the plane angle A' equals S/(3r^2), where S is the plane triangle's area.
On certain approximate Formul\ae
B = B' + \frac{S}{3r^2}Each spherical angle exceeds the corresponding plane angle by S/(3r^2); the same holds for B.
On certain approximate Formul\ae
A+B+C = A'+B'+C'+\frac{S}{r^2} = \pi + \frac{S}{r^2}The spherical excess (sum of the spherical angles minus pi) is approximately S/r^2, the plane area divided by the square of the radius.
On certain approximate Formul\ae
S = \tfrac{1}{2}\beta\gamma\sin A' = \tfrac{1}{2}\beta\gamma\sin AThe area of the plane triangle with two sides and the included angle, equal to the spherical one approximately.
On certain approximate Formul\ae
\sin B' = \frac{\beta}{\alpha}\sin A' = \frac{\beta}{\alpha}\sin AThe sine rule for the plane triangle, used to find B' from A, alpha and beta.
On certain approximate Formul\ae
S=\frac{\gamma^2 \sin A' \sin B'}{2\sin(A'+B')}The area of the plane triangle expressed from two angles and the included side gamma.
On certain approximate Formul\ae
S=\frac{\alpha^2\sin B' \sin C'}{2\sin(B'+C')}The area of the plane triangle expressed from two angles and the side alpha opposite one of them.
On certain approximate Formul\ae
A = A' + \frac{S}{3r^2}The spherical angle A equals the plane-triangle angle A' plus S/(3r^2).
On certain approximate Formul\ae
\sin\frac{1}{2}E = \frac{\sin\tfrac{1}{2}a \sin\tfrac{1}{2}b \sin C} {\cos\tfrac{1}{2}c}The sine of half the spherical excess E in terms of two sides and the included angle C.
On certain approximate Formul\ae
\sin C' \frac{\alpha\beta}{2r^2} \left( 1 + \frac{\alpha^2+\beta^2+\gamma^2}{24r^2} \right)A closer approximation to the spherical excess E, the plane-triangle sine term with a correction of order 1/r^2.
On certain approximate Formul\ae
\frac{\operatorname{Sin} A}{\operatorname{Sin} B} = \frac{\sin a}{\sin b}The sine rule for a spherical triangle: the ratio of the sines of two angles equals the ratio of the sines of the opposite sides.
On certain approximate Formul\ae
\frac{\alpha}{\beta} \left\{1 + \frac{\beta^2 - \alpha^2}{6r^2} \left(1 + \frac{7\beta^2-3\alpha^2}{60r^2}\right)\right\}Approximate value of sin A / sin B in terms of the arcs alpha and beta and the radius r.
On certain approximate Formul\ae
= \frac{\alpha^2-\beta^2}{\alpha\gamma\sin B} \left( 1 - \frac{\beta^2+\gamma^2-\alpha^2}{12r^2} \right)Approximate value of cot B minus cot A in terms of the sides and the radius.
On certain approximate Formul\ae
x = \alpha - \dfrac{\beta\sin A}{\sin B} - \dfrac{\mu (\alpha^2-\beta^2) }{\gamma\sin B}The error x in the calculated side alpha, when the side is found by the approximate formula with the spherical excess 3 mu.
On certain approximate Formul\ae
\mu=\dfrac{\alpha\gamma\sin B}{6r^2}Choice of the adopted spherical excess 3 mu that makes the error in the side vanish to the order r^-2, written from the Legendre formula.
On certain approximate Formul\ae
x=\frac{\alpha(\beta^2-\alpha^2)(3\alpha^2-7\beta^2)}{360r^4}The error in the calculated side when mu is taken from the Legendre formula, to order r^-4.
On certain approximate Formul\ae
= \frac{\alpha (\beta^2-\alpha^2) (\alpha^2 + \beta^2 - 5\gamma^2)} {720r^4}The error in the calculated side when mu is taken from an equation corresponding to (1) of Art. 109, to order r^-4.
Problems
Exercise IX
Exercise IX, problem 1, p. 093
If the sides of a spherical triangle $AB$, $AC$ be produced to $B'$, $C'$, so that $BB'$, $CC'$ are the semi-supplements of $AB$, $AC$ respectively, shew that the arc $B'C'$ will subtend an angle at the centre of the sphere equal to the angle between the chords of $AB$ and $AC$.
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Exercise IX, problem 10, p. 093
From Arts. 110 and 111, shew that approximately = + B - A + S3r^2(A-B).
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Exercise IX, problem 11, p. 093
By continuing the approximation in Art. 106 so as to include the terms involving $r^4$, shew that approximately A = A’ - ^2 A’6r^2 + (^2-3^2-3^2)^2 A’180r^4 .
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Exercise IX, problem 12, p. 093
From the preceding result shew that if $A = A' + \theta$ then approximately = A’6r^2 ( 1+7^2 + 7^2 + ^2120 r^2 ) .
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Exercise IX, problem 2, p. 093
Deduce Legendre’s Theorem from the formula ^2A2 = 12(a+b-c) 12(c+a-b) 12(b+c-a) 12(a+b+c) .
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Exercise IX, problem 3, p. 093
Four points $A$, $B$, $C$, $D$ on the surface of a sphere are joined by arcs of great circles, and $E$, $F$ are the middle points of the arcs $AC$, $BD$: shew that AB + BC + CD + DA = 4 AE BF FE.
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Exercise IX, problem 4, p. 093
If a quadrilateral $ABCD$ be inscribed in a small circle on a sphere so that two opposite angles $A$ and $C$ may be at opposite extremities of a diameter, the sum of the cosines of the sides is constant.
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Exercise IX, problem 5, p. 093
In a spherical triangle if $A = B = 2C$, shew that aa2 = ( c+a2).
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Exercise IX, problem 6, p. 093
$ABC$ is a spherical triangle each of whose sides is a quadrant; $P$ is any point within the triangle: shew that PA PB PC + BPC CPA APB = 0, and $ \tan ABP \tan BCP \tan CAP = 1. $and
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Exercise IX, problem 7, p. 093
If $O$ be the middle point of an equilateral triangle $ABC$, and $P$ any point on the surface of the sphere, then gather* 14 (PO OA)^2 (PA + PB + PC)^2 = ^2 PA + ^2 PB + ^2 PC - PA PB - PB PC - PC PA. gather*
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Exercise IX, problem 8, p. 093
If $ABC$ be a triangle having each side a quadrant, $O$ the pole of the inscribed circle, $P$ any point on the sphere, then (PA + PB + PC)^2 = 3^2 PO.
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Exercise IX, problem 9, p. 093
From each of three points on the surface of a sphere arcs are drawn on the surface to three other points situated on a great circle of the sphere, and their cosines are $a$, $b$, $c$; $a'$, $b'$, $c'$; $a''$, $b''$, $c''$. Shew that $ab''c' + a'bc'' + a''b'c = ab'c'' + a'b''c + a''bc'$.
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