Tiêu đề 44 Spherical Triangles and Polygons.pdf ✅

MSTC 44: Spherical Triangles and Polygons 1. Parts of a Sphere A sphere is a solid whose points on its surface are equidistant to a point called the center. The other parts of a sphere are: • Radius - The distance between the surface of a sphere and the center. • Chord - A segment whose endpoints are at the surface of the sphere. • Diameter - A chord passing through the center of the sphere. • Great Circle - A circle on the surface that contains the diameter. • Small Circle - A circle on the surface that does not contain the sphere's diameter. • Pole - The points of intersection of the axis of the sphere and the surface. • Arc - A segment along the surface of the sphere. These are measured by their angle subtended about the center. A spherical triangle is a figure on the surface of the sphere formed by three spherical arcs from a great circle. If more than three arcs form a figure, that is a spherical polygon. 2. Law of Cosines for Sides Consider a spherical triangle ABC centered at O, and a point P along radius OC. Let point Q be a point on sector AOB such that PQ is perpendicular to that sector, and points R and S on radii OA and OB, respectively, such that QR and QS are perpendicular to OQ. Let the arc subtended by spherical sides AB be c, BC be a, and AC be b.

Similarly, cos b = cos a cos c + sin a sin c cos B cos c = cos a cos b + sin a sin b cos C It is best to use the law of cosines for sides whenever two sides and the included angle are given. It can also be used when three sides are given. 3. Law of Cosines for Angles A polar triangle of a spherical triangle is another spherical triangle whose vertices are the poles of the great circles containing the sides of the original, and the vertices are in the same hemisphere as the vertices of the original. From the figure shown below, A′B′C′ is the polar triangle of ABC. A property of polar triangles is that all its sides are the supplements of the sides of the original, and all its angles are the supplements of the sides of the original. Using the law of cosines for sides on the polar triangle, cos a ′ = cos b ′ cos c ′ + sin b ′ sin c ′ cos A ′ cos(180° − A) = cos(180° − B) cos(180° − C) + sin(180° − B) sin(180° − C) cos(180° − a) For the supplements of angles, cos(180° − θ) = cos ⏞ 180° −1 cos θ − sin ⏞ 180° 0 sin θ cos(180° − θ) = − cos θ sin(180° − θ) = sin ⏞ 180° 0 cos θ − cos ⏞ 180° −1 sin θ sin(180° − θ) = sin θ Applying these identities, cos(180° − A) = cos(180° − B) cos(180° − C) + sin(180° − B) sin(180° − C) cos(180° − a) − cos A = (− cos B)(− cos C) + sin B sin C (− cos a) cos A = − cos B cos C + sin B sin C cos a
Similarly, cos B = − cos A cos C + sin Asin C cos b cos C = − cos A cos B + sin Asin B cos c It is best to use the law of cosines for angles whenever two angles and the included side are given. It can also be used when three angles are given. 4. Law of Sines From the spherical law of cosines for sides, cos a = cos b cos c + sin b sin c cos A sin b sin c cos A = cos a − cos b cos c Squaring both sides and applying Pythagorean identities, sin2 b sin2 c cos2 A = cos2 a − 2 cos a cos b cos c + cos2 b cos2 c sin2 b sin2 c sin2 A = 1 − cos2 a − cos2 b − cos2 c + 2 cos a cos b cos c Dividing both sides by sin2 a sin2 b sin2 c, sin2 A sin2 a = 1 − cos2 a − cos2 b − cos2 c + 2 cos a cos b cos c sin2 a sin2 b sin2 c Following the same procedure from the other equations of the law of cosines, sin2 B sin2 b = 1 − cos2 a − cos2 b − cos2 c + 2 cos a cos b cos c sin2 a sin2 b sin2 c sin2 C sin2 c = 1 − cos2 a − cos2 b − cos2 c + 2 cos a cos b cos c sin2 a sin2 b sin2 c Thus, sin2 A sin2 a = sin2 B sin2 b = sin2 C sin2 c sin A sin a = sin B sin b = sin C sin c It is good to use the law of sines when there is a given pair of angle and its opposite side. However, due to the supplementary identity sin(180° − θ) = sin θ, the solver must always consider two answers. 5. Napier’s Rules A right spherical triangle is a triangle where one of its angles is 90°. Thus, the laws of sines and cosines simplify since a value is already known. The right angle is commonly placed at C. A visual mnemonic used to remember the rules is shown, where an overbar denotes the complement of that angle.