Spherical Trigonometry, for the Use of Colleges and Schools

On the connexion of Formul\ae\ in Plane and Spherical Trigonometry

Excerpts

On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
then if we suppose $r$ to become indefinitely great, the limiting form of the proposed formula will be a relation in Plane Trigonometry.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
that is, in a plane triangle the sides are as the sines of the opposite angles.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Let $OS = \alpha$, $OSX = \beta$; then the position of $O$ is determined by means of these angular co-ordinates $\alpha$ and $\beta$.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
It will be observed that the angular co-ordinates here used are analogous to the *latitude* and *longitude* which serve to determine the positions of places on the Earth’s surface;
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
It is known in Plane Geometry that a certain circle touches the inscribed and escribed circles of any triangle; this circle is called the *Nine points circle*:
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
The principal use of Art. 137 is to determine whether three given points are on the same great circle; an illustration will be given in Art. 146.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
The arcs drawn from the angles of a spherical triangle perpendicular to the opposite sides respectively meet at a point.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
The student must have perceived that many of the results obtained in *Spherical* Trigonometry resemble others with which he is familiar in *Plane* Trigonometry.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
if we suppose $r$ to become indefinitely great, the limiting form of the proposed formula will be a relation in Plane Trigonometry.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Let $O$ be the pole of a small circle, $S$ a fixed point on the sphere, $SX$ a fixed great circle of the sphere.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
this gives a relation between the angular co-ordinates of any point on the circumference of the circle.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
this result corresponds to the well-known property of a circle in Plane Geometry which is demonstrated in Euclid iii. 36 *Corollary*.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
We shall now shew that a small circle can always be determined on the sphere to touch the inscribed and escribed circles of any spherical triangle.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
The student should convince himself by examination that the result holds for all relative positions of $P$, $P_1$ and $P_2$, when due regard is paid to algebraical signs.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
The results which have been demonstrated with respect to the circle which touches the inscribed and escribed circles of a spherical triangle are mainly due to Dr Hart and Dr Salmon.

Equations

On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos(\rho+r_1) = \cos\alpha_1\cos\beta + \sin\alpha_1\sin\beta\cos\gamma
Condition that the touching circle touches the escribed circle of angular radius r1 externally.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos A = \frac{\cos a - \cos b \cos c }{\sin b \sin c}
Spherical cosine rule for angle A, quoted as the starting formula from which the plane result is deduced by letting the sphere's radius grow without limit.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos A = \frac{\beta^2 + \gamma^2 - \alpha^2}{2 \beta \gamma}\,
Cosine of an angle of a plane triangle in terms of its three sides, obtained as the limit of the spherical formula when the radius becomes infinite.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\sin A}{\sin B} = \frac{\sin a}{\sin b}
On a sphere the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles; quoted as the starting formula.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\sin A}{\sin B} = \frac{\alpha}{\beta}
In a plane triangle the sides are as the sines of the opposite angles, the limiting form of the spherical law as the radius becomes infinite.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos r = \cos \alpha \cos \theta + \sin \alpha \sin \theta \cos (\phi - \beta)
Equation of a small circle of angular radius r on the sphere, in angular co-ordinates of its pole and of a general point of the circle.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
0 = \cos \alpha \cos \theta + \sin \alpha \sin \theta \cos (\phi - \beta)
Equation of a great circle on the sphere, the special case r = pi/2 of the small circle equation.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\tan \frac{\theta_1}{2} \tan \frac{\theta_2}{2}= \frac{\cos r - \cos \alpha}{\cos r + \cos \alpha}
The product of the two roots of the quadratic in tan(theta/2) is independent of phi, the spherical analogue of an Euclid III.36 property.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\cos r - \cos \alpha}{\cos r + \cos \alpha} = \tan \frac{\alpha + r}{2} \tan \frac{\alpha - r}{2}
The cosine ratio equals the product of tangents of half-sums and half-differences of alpha and r.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\sin PM = \sin OP \sin AOB
Perpendicular from P to the arc OA is found from the sine of OP times the sine of the angle AOB (right-angled triangle relation, Art. 65).
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\sin PM}{\sin PN} = \frac{\sin AOB}{\sin COB}
The ratio of the sines of the perpendiculars from any point P on OB is independent of P's position.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\sin x = \frac{\sin \theta_2}{\sin(\theta_1 + \theta_2)} \sin x_1 + \frac{\sin \theta_1}{\sin(\theta_1 + \theta_2)} \sin x_2
Three-point relation: the sine of the perpendicular from P is a weighted sum of the sines of the perpendiculars from P1 and P2 to the fixed arc.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos B = \cos CF \sin FCB
Cosine of an angle of the triangle equals the cosine of the perpendicular CF times the sine of the angle FCB (Art. 65).
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\sin x}{\cos B \cos C} = \frac{\sin y}{\cos C \cos A} = \frac{\sin z}{\cos A \cos B}
The three perpendiculars from the point of concurrence of the altitudes, divided by the products of cosines of the angles, are equal; this fixes the point where the perpendiculars meet.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\sin x}{\sin B \sin C} = \frac{\sin y}{\sin C \sin A} = \frac{\sin z}{\sin A \sin B}
For the point where the arcs to the midpoints of the opposite sides meet, the perpendiculars divided by sine-products of the angles are equal.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos(\rho-r) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\cos\gamma
Condition that the touching circle of angular radius rho touches the inscribed circle internally.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos(\rho+r_2) = \cos\alpha_2\cos\beta + \sin\alpha_2\sin\beta \cos\left(\frac{\pi}{2}-\gamma\right)
Condition that the touching circle touches the second escribed circle externally, with the angle replaced by pi/2 minus gamma.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos(\rho+r_3) = \cos\alpha_3\cos\beta + \sin\alpha_3\sin\beta \cos\left(\frac{\pi}{2}+\gamma\right)
Condition that the touching circle touches the third escribed circle externally, with the angle replaced by pi/2 plus gamma.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
2\cos\rho \sin\frac{a}{2}\cos\frac{b+c}{2} + 2n\sin\rho=\cos\beta\sin a
Elimination of cos gamma between the first two touching conditions gives this relation between rho, beta and the sides.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
2\cos\rho\sin\frac{a}{2}\cos\frac{b-c}{2}-2n\sin\rho=\cos\beta\sin a
Companion relation from the elimination of sin gamma, with b - c in place of b + c and the sign of the n term reversed.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\tan\rho= \frac{\sin\dfrac{a}{2}\sin\dfrac{b}{2}\sin\dfrac{c}{2}}{n} = \frac{1}{2}\tan R
The angular radius of the touching circle is determined by tan rho, which is half tan R (the circumradius relation of Art. 92).
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\cos\beta= \dfrac{\cos\dfrac{b}{2}\cos\dfrac{c}{2}\cos\rho}{\cos\dfrac{a}{2}}
The distance beta of the touching circle's pole from A is fixed by cos beta.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\tan\frac{\lambda}{2} \tan\frac{\mu}{2} = \frac{\cos\rho - \cos\beta}{\cos\rho + \cos\beta}
Product of the tangents of half the distances from A to the two points where the touching circle meets side AB (Art. 134 applied).
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\tan\frac{\lambda}{2} = \frac{\cos \dfrac{a}{2} - \cos \dfrac{b}{2} \cos \dfrac{c}{2} } {\cos \dfrac{b}{2} \sin \dfrac{c}{2} }
Tangent of half the distance lambda from A to the first intersection point of the touching circle with AB.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\tan\frac{\mu}{2} = \frac{\cos \dfrac{b}{2} \sin \dfrac{c}{2} } {\cos \dfrac{a}{2} + \cos \dfrac{b}{2} \cos \dfrac{c}{2} }
Tangent of half the distance mu from A to the second intersection point of the touching circle with AB.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\sin z = \frac{\cos \rho}{n} \sin\frac{a}{2} \sin\frac{b}{2} \sin\frac{c}{2} \cos(A - B)
The perpendicular z from the pole of the touching circle to side AB equals sin rho times cos(A - B), written in terms of the sides.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\frac{\sin x}{\cos(B - C)} = \frac{\sin y}{\cos(C - A)} = \frac{\sin z}{\cos(A - B)}
Perpendiculars from the pole of the touching circle, divided by cosines of angle differences, are equal.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
z = \dfrac12 R \cos(A - B)
Plane-geometry limit of the perpendicular to the sides: the Nine points circle property, obtained when the sphere's radius becomes infinite.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\lambda = \dfrac{b^2 + c^2 - a^2}{2c}
Plane limit of equation (4): the nine-points circle passes through the feet of the altitudes.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\mu = \dfrac{c}{2}
Plane limit of equation (5): the nine-points circle passes through the midpoints of the sides.
On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
\sin z = \sin \beta \sin \left(\frac{A}{2} + \gamma\right)
The perpendicular z from the pole of the touching circle to AB, written through the angles beta and gamma (Art. 145).

Problems

Exercise XII

Exercise XII, problem 1, p. 122
From the formula $\sin\dfrac{a}{2}=\Surd{\left\{\dfrac{-\cos S\cos(S-A)}{\sin B\sin C}\right\}}$ deduce the expression for the area of a plane triangle, namely $\dfrac{a^2\sin B\sin C}{2\sin A}$, when the radius of the sphere is indefinitely increased.
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Exercise XII, problem 10, p. 122
Shew that the points determined in Examples 8 and 9, and the point $N$ of Art. 146 are on a great circle. State the corresponding theorem in Plane Geometry.
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Exercise XII, problem 11, p. 122
If one angle of a spherical triangle remains constant while the adjacent sides are increased, shew that the area and the sum of the angles are increased.
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Exercise XII, problem 12, p. 122
If the arcs bisecting two angles of a spherical triangle and terminated at the opposite sides are equal, the bisected angles will be equal provided their sum be less than $180^\circ$.
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Exercise XII, problem 2, p. 122
Two triangles $ABC$, $abc$, spherical or plane, equal in all respects, differ slightly in position: shew that ABbBCcCAa+ACcCBbBAa=0.
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Exercise XII, problem 3, p. 122
Deduce formul in Plane Trigonometry from Napier’s Analogies.
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Exercise XII, problem 4, p. 122
Deduce formul in Plane Trigonometry from Delambre’s Analogies.
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Exercise XII, problem 5, p. 122
From the formula $\cos\dfrac{c}{2}\cos\dfrac{A+B}{2} =\sin\dfrac{C}{2}\cos\dfrac{a+b}{2}$ deduce the area of a plane triangle in terms of the sides and one of the angles.
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Exercise XII, problem 6, p. 122
What result is obtained from Example 7 to Chapter VI., by supposing the radius of the sphere infinite?
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Exercise XII, problem 7, p. 122
From the angle $C$ of a spherical triangle a perpendicular is drawn to the arc which joins the middle points of the sides $a$ and $b$: shew that this perpendicular makes an angle $S-B$ with the side $a$, and an angle $S-A$ with the side $b$.
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Exercise XII, problem 8, p. 122
From each angle of a spherical triangle a perpendicular is drawn to the arc which joins the middle points of the adjacent sides. Shew that these perpendiculars meet at a point; and that %-----File: 123.png------------------------------------------------ if $x$, $y$, $z$ are the perpendiculars from this point on the sides $a$, $b$, $c$ respectively, x(S-B)(S-C) = y(S-C)(S-A) = z(S-A)(S-B).
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Exercise XII, problem 9, p. 122
Through each angle of a spherical triangle an arc is drawn so as to make the same angle with one side which the perpendicular on the base makes with the other side. Shew that these arcs meet at a point; and that if $x$, $y$, $z$ are the perpendiculars from this point on the sides $a$, $b$, $c$ respectively, xA=yB=zC.
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