Question

Al-T?si demonstrated a method to solve a spherical triangle if all three angles are known. Suppose the three angles of triangle \(A B C\) are given (Fig. 9.39), where we assume that all three sides of the triangle are less than a quadrant. We extend each side of the triangle two different ways to form a quadrant. That is, we extend \(A B\) to \(A D\) and \(B H\); \(A C\). to \(A E\) and \(C G\); and \(B C\) to \(B K\) and \(F C\), where all of the six new arcs are quadrants. We then draw great circle arcs through \(D\) and \(E, F\) and \(G\), and \(H\) and \(K\) to form the new spherical triangle \(L M N\). Now the vertices of the original triangle are the poles of the three sides of the new triangle. Then, for example, \(M D=E N=90^{\circ}-D E=90^{\circ}-A\), or \(M N=180^{\circ}-A\). Thus, the three sides of triangle \(L M N\) are known, and therefore the triangle can be solved by the procedure sketched in the text. But we also know that the vertices of triangle \(L M N\) are the poles of the original triangle. So, for example, \(B F=C K=90^{\circ}-B C\), and \(L=F K=180^{\circ}-B C\). We therefore can determine the sides of the original triangle. Use this procedure to solve the triangle \(A B C\), where \(A=75^{\circ}, B=80^{\circ}\), and \(C=85^{\circ}\).

Short answer

Question: Using Al-Tusi's method, find the sides of a spherical triangle ABC, given that the angles A, B, and C are 75°, 80°, and 85°, respectively. Answer: The sides of the spherical triangle ABC are: - AB = 75° - AC = 85° - BC = 68.86°

Step-by-step solution

Extend the Triangle ABC and Form Triangle LMN

To form the new triangle LMN, we extend each side of the triangle ABC to form quadrant arcs: - Extend AB to AD and BH; - Extend AC to AE and CG; - Extend BC to BK and FC. Now, we draw great circle arcs through D and E, F, and G, H, and K to form the new spherical triangle LMN. We know that the vertices of triangle ABC are the poles of the three sides of the new triangle LMN.

Calculate Sides of Triangle LMN

Using the angles of the original triangle ABC, we can calculate the sides of triangle LMN as follows: - MN = 180° - A = 180° - 75° = 105° - LN = 180° - B = 180° - 80° = 100° - LM = 180° - C = 180° - 85° = 95°

Solve Triangle LMN

To solve triangle LMN, we use the Spherical Law of Cosines for sides: - cos(LM) = cos(LN) * cos(MN) + sin(LN) * sin(MN) * cos(L) To find angle L: - cos(L) = (cos(LM) - cos(LN) * cos(MN)) / (sin(LN) * sin(MN)) - cos(L) = (cos(95°) - cos(100°) * cos(105°)) / (sin(100°) * sin(105°)) - L = arccos(cos(L)) ≈ 111.14°

Calculate Sides of Original Triangle ABC

Now that we have found angle L, we can find the sides of the original triangle ABC using the vertices of triangle LMN as poles: - BC = 180° - L = 180° - 111.14° ≈ 68.86° - AC = 180° - M = 180° - 95° = 85° - AB = 180° - N = 180° - 105° = 75° Therefore, the sides of the original triangle ABC are as follows: - AB = 75° - AC = 85° - BC = 68.86°

Key concepts

Spherical Geometry

Spherical geometry is a type of non-Euclidean geometry that deals with figures on the surface of a sphere, as opposed to the flat planes of classical geometry. Unlike Euclidean geometry, where the sum of the angles in a triangle equals 180 degrees, in spherical geometry the sum of angles is greater than 180 degrees. Spherical triangles, formed by three arcs of great circles, are the fundamental shapes studied in this geometry.

Students studying spherical geometry should consider that this system has different rules due to the curvature of the sphere. For example, the shortest distance between two points is an arc of a great circle (analogous to a straight line in flat geometry). Also, there can be no parallel lines since all great circles eventually intersect. These unique attributes are essential when considering problems involving the Earth's surface or celestial navigation.

Spherical Triangle Properties

Spherical triangles are bounded by arcs of great circles on the surface of a sphere. The properties of these triangles differ significantly from those of their flat counterparts. A notable property is the angle sum of a spherical triangle, which exceeds 180 degrees and can be up to 540 degrees.

Another interesting aspect is that each angle of a spherical triangle is larger than the difference between 180 degrees and the opposite side (measured in degrees). Also, the sides of spherical triangles are measured by their angular distance, typically in degrees, rather than linear distance. When solving for a spherical triangle, this calls for using trigonometric functions adapted for use on a sphere, such as the Spherical Law of Cosines.

Spherical Law of Cosines

The Spherical Law of Cosines is a crucial theorem for solving spherical triangles, analogous to the Law of Cosines in plane geometry. The law relates the sides and angles of a spherical triangle through the cosine function. The theorem states:

\[\cos(a) = \cos(b) \cdot \cos(c) + \sin(b) \cdot \sin(c) \cdot \cos(A),\]

where \(a\), \(b\), and \(c\) are the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively.

Applying this to the problem given in Al-Tūsī's method, we first determine the sides of the auxiliary triangle LMN by subtracting the known angles from 180 degrees. Then, we use the Spherical Law of Cosines to solve for the remaining angles of LMN, which yields the lengths of the original triangle's sides. This law is powerful because it connects spherical angles and arcs in a way that can be used to solve complex navigation and astronomy problems where spherical geometry is applicable.