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# Spherical Trigonometry

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SPHERICAL TRIGONOMETRY DEFINITION OF TERMS The sphere is the set of all points in a three-dimensional space such that the distance of each from a fixed point is constant. The fixed point and the given distance are called the center and the radius of the sphere respectively. The intersection of a plane with a sphere is a circle. If the plane passes through the center of the sphere, the intersection is a great circle; otherwise, the intersection is a small circle.

A line perpendicular to the plane of a circle and through the center of the sphere is called the axis of the circle.

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The intersection of this axis and the sphere are called the poles of the circle. Opposite ends of a diameter are identified as antipodal points. Two great circles intersecting in a pair of antipodal points divide the sphere into four regions called lunes. Thus a lune is bounded by the arcs of two great circles. The polar distance (in angular units) of a circle is the least distance of a point on the circle to its pole.

Two distinct points on the sphere which are not ends of a diameter divide the great circle into two arcs. The shorter arc is called the minor arc. SPHERICAL TRIANGLES

A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersections of these arcs are called the vertices of the spherical triangle. The angle formed by two intersecting arcs is called a spherical angle. Like a plane triangle, the spherical triangle has also six parts – three angles and three sides. The sides a, b, and c are measured by the corresponding faces of the trihedral angle. Important Propositions from Solid Geometry: 1. If two sides are equal, the angles opposite are equal and conversely. . If two sides are unequal, the angles opposite are unequal and the greater side is opposite the greater angle and conversely. 3. The sum of any two sides is greater than the third side, that is, a + b &gt; c, a + c &gt; b, b + c &gt; a 4. The sum of the three sides is less than 3600, that is 00 &lt; a + b + c &lt; 3600 5. The sum of any two angles is less than 1800 plus the third angle, that is, A + B &lt; 1800 + C, A + C &lt; 1800 + B, B + C &lt; 1800 + A 6. The sum of the angles is greater than 1800 and less than 5400, that is 1800 &lt; A + B + C &lt; 5400.

Example 1. a. The angles of a spherical triangle are A = 520, B = 510 and C = 900. The sides are 600, 430, and 470. Which side is c? b. If a = 350, b = 700 and c = 1150, is the spherical triangle with these sides possible? RIGHT SPHERICAL TRIANGLES Fundamental Formulas sin a = sin c sin Atan b = tan c cos A sin b = sin c sin Bcos c = cos a cos b tan a = sin b tan Acos c = cot A cot B tan b = sin a tan Bcos A = cos a sin B tan a = tan c cos Bcos B = cos b sin A Laws of Quadrants LQ1: Any side and its opposite angle lie in the same quadrant and conversely.

LQ2: (a) If any two sides lie in the same quadrant, then the third side is less than 90 and0 conversely. (b) If any two sides lie in different quadrants, then the third side is greater than 900 and conversely. Example 2. Answer the following by using the symbol &lt; or &gt;. a. If a &lt; 900, what is the value of A? b. If b &gt; 900, what is the value of B? c. If a &gt; 900, and b &gt; 900 what is the value of c? d. If a &lt; 900, and b &lt; 900 what is the value of c? e. If a &lt; 900, and b &gt; 900 what is the value of c? f. If a &lt; 900, and b &lt; 900 what are the values of A and B? g. If c &gt; 900, what are the values of a and b? . If c &lt; 900, what are the values of A and B? Napier’s Rules NR1: The sine of any middle part is equal to the product of the tangents of the adjacent parts. NR2: The sine of any middle part is equal to the product of the cosines of the opposite parts. Example 3. Apply Napier’s rules to find the formula for each of the unknown parts given the following parts. a. a, b b. c, a c. A, a d. B, a e. A, B Solutions of Right Spherical Triangles To solve a right spherical triangle having two given parts, the following steps may be used: Step 1. Draw a schematic diagram which exhibits the circular parts and then encircle the parts given.

Step 2. Apply Napier’s rules using the figure in step 1 to obtain the necessary formula. Step 3. Compute the values of the unknown part and apply the laws of quadrants. Example 4. Solve the right spherical triangle (C = 900) given a. b = 48030’, c = 69040’ b. c = 720, A = 1560 c. b = 36010’, B = 52040’ Quadrantal and Isosceles Spherical Triangles A quadrantal triangle is a spherical triangle having a side equal to 900. It is not necessarily a right spherical triangle. If ABC is a quadrantal triangle, its polar triangle A’B’C’ is a right spherical triangle. The solutions of A’B’C’ yields the solution of ABC.

An isosceles spherical triangle (not necessarily a right triangle) is a spherical triangle with at least two equal sides. Example 5. a. Solve the quadrantal triancle (c = 900), given A = 1150, b = 1400. b. Solve the sphrerical triangle with A = C = 640 and b = 820. OBLIQUE SPHERICAL TRIANGLES The Six Cases Case I: Given three sides Case II: Given three angles Case III: Given two sides and the included angle Case IV: Given two angles and the included side Case V: Given two sides and the angle opposite one of them Case VI: Given two angels and the side opposite one of them The Law of Sines The Law of Cosines for Sides

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