Spherical Trigonometry, for the Use of Colleges and Schools
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
Excerpts
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
The radical on the right-hand side must be taken with the positive sign, because $\sin b$, $\sin c$, and $\sin A$ are all positive.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
It should be observed that the two triangles in this case are *not* necessarily such that one may be made to *coincide with the other by superposition*.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
The formul (4), (5), (6), (7) may be put in the form of proportions or analogies, and are called from their discoverer *Napier’s Analogies:* the last two may be demonstrated without recurring to the polar triangle by starting with the formul in Art. 39.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
The last four formul are commonly, but improperly, called *Gauss’s Theorems*; they were first given by Delambre in the *Connaissance des Tems* for 1809, page 445.
Equations
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos a = \cos b \cos c + \sin b \sin c \cos AThe cosine of a side equals the product of the cosines of the other two sides plus the product of their sines times the cosine of the included angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos A = \dfrac{\cos a - \cos b \cos c}{\sin b \sin c}The cosine of an angle of a spherical triangle is expressed in terms of the cosines and sines of the sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos b = \cos c \cos a + \sin c \sin a \cos BThe cosine of side b equals the cosine of c times the cosine of a plus the sines of c and a times the cosine of the angle B.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos c = \cos a \cos b + \sin a \sin b \cos CThe cosine of side c equals the cosine of a times the cosine of b plus the sines of a and b times the cosine of the angle C.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos a = \sin b \cos AIn the case where one side containing the angle A is a quadrant, the cosine of the opposite side equals the sine of b times the cosine of A.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin A=\dfrac{\surd(1-\cos^2 a-\cos^2 b-\cos^2 c+2\cos a\cos b\cos c)}{\sin b \sin c}The sine of an angle of a spherical triangle is given in terms of the trigonometrical functions of the sides, with the positive root taken.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\dfrac{\sin A}{\sin a}=\dfrac{\sin B}{\sin b}=\dfrac{\sin C}{\sin c}The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\dfrac{\sin B}{\sin C}=\dfrac{\sin b}{\sin c}The sines of two angles of a spherical triangle are in the same ratio as the sines of the opposite sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cot a \sin b = \cot A \sin C + \cos b \cos CA four-part relation between two sides and two angles of a spherical triangle, involving cotangents and sines of one side and cosines of the other.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin^2 \dfrac{A}{2} = \dfrac{\sin \tfrac{1}{2}(a+b-c)\sin\tfrac{1}{2}(a-b+c)}{\sin b \sin c}The square of the sine of half an angle of a spherical triangle is expressed as a product of sines of half-sums of the sides over the product of the sines of the two sides that contain the angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin^2 \dfrac{A}{2} = \dfrac{\sin(s - b)\sin(s - c)}{\sin b \sin c}The square of the sine of half an angle equals the product of the sines of s minus each adjacent side over the product of the sines of the two sides containing the angle, where 2s is the sum of the sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\tan\dfrac{A}{2}=\Surd{\left\{ \dfrac{\sin(s-b)\sin(s-c)}{\sin s \sin(s-a)} \right\}}The tangent of half an angle of a spherical triangle is the square root of a ratio of sines of s minus the sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin A = \dfrac{2}{\sin b \sin c}\{\sin s \sin(s-a)\sin(s-b)\sin(s-c)\}^{\tfrac{1}{2}}The sine of an angle of a spherical triangle expressed as twice the square root of the product of four sines of s and s minus the sides, over the product of the sines of the two adjacent sides.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos A =-\cos B \cos C + \sin B \sin C \cos aThe cosine of an angle of a spherical triangle expressed in terms of the cosines and sines of the other two angles and the opposite side (the polar form of the cosine rule).
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin^{2}\frac{a}{2}=-\frac{\cos S\cos(S-A)}{\sin B\sin C}The square of the sine of half a side is expressed in terms of cosines of S and S minus A over the product of the sines of the two angles adjacent to the side, where 2S is the sum of the angles.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin a=\dfrac{2}{\sin B\sin C}\left\{-\cos S\cos (S-A)\cos(S-B)\cos (S-C)\right\}^{\tfrac{1}{2}}The sine of a side is twice the square root of minus the product of four cosines of S and S minus the angles, over the product of the sines of the two adjacent angles.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\tan\tfrac{1}{2}(A + B) = \frac{\cos\tfrac{1}{2}(a - b)}{\cos\tfrac{1}{2}(a + b)}\cot\frac{C}{2}Napier's first analogy: the tangent of half the sum of two angles equals the ratio of cosines of half the difference and half the sum of the opposite sides, times the cotangent of half the third angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\tan\frac{1}{2}(A - B) = \frac{\sin\tfrac{1}{2}(a - b)} {\sin\tfrac{1}{2}(a + b)} \cot\frac{C}{2}Napier's second analogy: the tangent of half the difference of two angles equals the ratio of sines of half the difference and half the sum of the opposite sides, times the cotangent of half the third angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\tan\tfrac{1}{2}(a + b) = \frac{\cos\tfrac{1}{2}(A - B)} {\cos\tfrac{1}{2}(A + B)} \tan\frac{c}{2}Napier's third analogy, obtained from the first by the supplemental triangle: the tangent of half the sum of two sides equals the ratio of cosines of half the difference and half the sum of the opposite angles, times the tangent of half the third side.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\tan\tfrac{1}{2}(a - b) = \frac{\sin\tfrac{1}{2}(A - B)} {\sin\tfrac{1}{2}(A + B)} \tan\frac{c}{2}Napier's fourth analogy: the tangent of half the difference of two sides equals the ratio of sines of half the difference and half the sum of the opposite angles, times the tangent of half the third side.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos^2\tfrac{1}{2}c = \cos^2\tfrac{1}{2}(a - b) \cos^2\tfrac{1}{2}C + \cos^2\tfrac{1}{2}(a + b) \sin^2\tfrac{1}{2}CThe square of the cosine of half the side c is expressed as a weighted sum of the squares of cosines of half the difference and half the sum of a and b, weighted by the squared cosine and sine of half the angle C.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos\tfrac{1}{2}(A + B) \cos\tfrac{1}{2}c = \cos\tfrac{1}{2}(a + b) \sin\tfrac{1}{2}CDelambre's relation between the cosines of half the sum of two angles and half the third side, and the cosine of half the sum of the sides and the sine of half the third angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos\tfrac{1}{2}(A - B) \sin\tfrac{1}{2}c = \sin\tfrac{1}{2}(a + b) \sin\tfrac{1}{2}CDelambre's relation between the cosine of half the difference of two angles times the sine of half the third side, and the sine of half the sum of the two sides times the sine of half the third angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin\tfrac{1}{2}(A + B) \cos\tfrac{1}{2}c = \cos\tfrac{1}{2}(a - b) \cos\tfrac{1}{2}CThe sine of half the sum of two angles times the cosine of half the third side equals the cosine of half the difference of the sides times the cosine of half the third angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\sin\tfrac{1}{2}(A - B) \sin\tfrac{1}{2}c = \sin\tfrac{1}{2}(a - b) \cos\tfrac{1}{2}CThe sine of half the difference of two angles times the sine of half the third side equals the sine of half the difference of the sides times the cosine of half the third angle.
Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
\cos a \cos b + \sin a \sin b \cos C = \cos cThe fundamental cosine relation for the sides, derived by the coordinate method with the origin at the centre of the sphere.
Problems
Exercise IV
Exercise IV, problem 1, p. 040
If $A = a$, shew that $B$ and $b$ are equal or supplemental, as also $C$ and $c$.
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Exercise IV, problem 10, p. 040
$AB$, $CD$ are quadrants on the surface of a sphere intersecting at $E$, the extremities being joined by great circles: shew that AEC = AC BD - BC AD.
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Exercise IV, problem 11, p. 040
If $b + c = \pi$, shew that $\sin 2B + \sin 2C = 0$.
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Exercise IV, problem 12, p. 040
If $DE$ be an arc of a great circle bisecting the sides $AB$, $AC$ of a spherical triangle at $D$ and $E$, $P$ a pole of $DE$, and $PB$, $PD$, $PE$, $PC$ be joined by arcs of great circles, shew that the angle $BPC =$ twice the angle $DPE$.
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Exercise IV, problem 13, p. 040
In a spherical triangle shew that b c + b c A = B C - B C a.
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Exercise IV, problem 14, p. 040
If $D$ be any point in the side $BC$ of a triangle, shew that AD BC = AB DC + AC BD.
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Exercise IV, problem 15, p. 040
In a spherical triangle shew that $\theta$, $\phi$, $\psi$ be the lengths of arcs of great circles drawn from $A$, $B$, $C$ perpendicular to the opposite sides, gather* a = b = c = (1 - ^2 a - ^2 b - ^2 c + 2 a b c). gather*
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Exercise IV, problem 16, p. 040
In a spherical triangle, if $\theta$, $\phi$, $\psi$ be the arcs bisecting the angles $A$, $B$, $C$ respectively and terminated by the opposite sides, shew that A2 + B2 + C2 = a + b + c.
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Exercise IV, problem 17, p. 040
Two ports are in the same parallel of latitude, their common latitude being $l$ and their difference of longitude $2\lambda$: shew that the saving of distance in sailing from one to the other on the great circle, instead of sailing due East or West, is 2rl - ^-1(l), $\lambda$ being expressed in circular measure, and $r$ being the radius of the Earth.
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Exercise IV, problem 18, p. 040
If a ship be proceeding uniformly along a great circle and the observed latitudes be $l_1$, $l_2$, $l_3$, at equal intervals of time, in each of which the distance traversed is $s$, shew that s = r^-1 12(l_1 + l_3)12(l_1 - l_3) l_2, $r$ denoting the Earth’s radius: and shew that the change of longitude may also be found in terms of the three latitudes.
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Exercise IV, problem 2, p. 040
If one angle of a triangle be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.
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Exercise IV, problem 3, p. 040
When does the polar triangle coincide with the primitive triangle?
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Exercise IV, problem 4, p. 040
If $D$ be the middle point of $AB$, shew that AC + BC = 2 12 AB CD.
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Exercise IV, problem 5, p. 040
If two angles of a spherical triangle be respectively equal to the sides opposite to them, shew that the remaining side is the supplement of the remaining angle; or else that the triangle has two quadrants and two right angles, and then the remaining side is equal to the remaining angle.
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Exercise IV, problem 6, p. 040
In an equilateral triangle, shew that $2 \cos \dfrac a2 \sin \dfrac A2 = 1$.
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Exercise IV, problem 7, p. 040
In an equilateral triangle, shew that $\tan^2 \dfrac a2 = 1 - 2 \cos A$; hence deduce the limits between which the sides and the angles of an equilateral triangle are restricted.
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Exercise IV, problem 8, p. 040
In an equilateral triangle, shew that $\sec A = 1 + \sec a$.
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Exercise IV, problem 9, p. 040
If the three sides of a spherical triangle be halved and a new triangle formed, the angle $\theta$ between the new sides $\dfrac b2$ and $\dfrac c2$ is given by $\cos \theta = \cos A + \tfrac 12 \tan \dfrac b2 \tan \dfrac c2 \sin^2 \theta$.
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