Math12 lesson10
- 2. DEFINITION OF TERMS:Spherical Trigonometry – is a branch of trigonometry that concerns with triangles extracted from the surface of the sphere.
- 3. Great Circle – is a circle obtained by passing a section through the center of the sphere.
- 4. Spherical Triangle – is a spherical surface bounded by the area of three great circles.
- 5. Right Spherical Triangle – is a spherical triangle having a right angle. SPHERICAL TRIANGLEA 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 intersection of these arcs are called the vertices. The angle formed by two intersecting arcs is called a spherical angle.ACAbcCBOOaB
- 6. Important Propositions from Solid GeometryIf two sides are equal, the angles opposite are equal and conversely.
- 7. If two sides are unequal , the angles opposite are unequal and the greater side is opposite the greater angle and conversely.
- 8. The sum of any two sides is greater than the third side, that is, a + b > c, a + c > b, b + c > aThe sum of the three sides is less than 3600, that is, 00 < a + b + c < 3600The sum of any two angles is less than 1800 plus the third angle, that is, A + B < 1800 + C, A + C < 1800 + B, B + C < 1800 + AThe sum of the angles is greater than 1800 and less than 5400, that is, 1800 < A + B + C < 5400
- 9. RIGHT SPHERICAL TRIANGLENapier’s CircleBco - Bco - ccaaco - AbWhere: co – A = complement of A co – B = complement of B co – c = complement of c ACb
- 10. NAPIER’S RULENR1: 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.Laws of QuadrantsLQ1: 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 900 and conversely. (b) If any two sides lie in different quadrants, then the third side is greater than 900 and conversely.
- 11. FUNDAMENTAL FORMULAS:tan a = tan c cos B
- 12. tan a = sin b tan A
- 13. tan b = sin a tan B
- 14. tan b = tan c cos A
- 15. cos c = cot A cot B
- 16. sin a = sin c sin A
- 17. sin b = sin c sin B
- 18. cos c = cos a cos b
- 19. cos A = cos a sin B
- 20. cos B = cos b sin AEXAMPLE:Solve the following right spherical triangle: (C = 900)a = 50020’ c = 80030’ a = 45057’ b = 68027’B = 52037’ b = 48020’B = 48025’ b = 52017’a = 98052’ A = 38014’
- 21. OBLIQUE SPHERICAL TRIANGLELAW OF SINES:LAW OF COSINES: ( for the sides )cos a = cos b cos c + sin b sin c cos Acos b = cos a cos c + sin a sin c cos Bcos c = cos a cos b + sin a sin b cos CLAW OF COSINES: ( for the angles )cos A = - cos B cos C + sin B sin C cos acos B = - cos A cos C + sin A sin C cos bcos C = - cos A cos B + sin A sin B cos c
- 22. NAPIER’S ANALOGIES:TERRESTRIAL SPHERE: where: E = spherical excess in degrees E = ( A+B+C) - 180
- 23. Solution of oblique spherical triangle involves six cases, namely:Case 1: Two sides and included angle are given.Case 2: Two angles and the included side.Case 3: Two sides and an angle opposite one of them.Case 4: Two angles and a side opposite one of them.Case 5: Three sides are given.Case 6: Three angles are given.
- 24. EXAMPLE:Solve the following triangles.a = 750 b = 1100 C = 720c = 1130 B = 660 A = 1100a = 540 b = 250 A = 1010a = 250 B = 1080 A = 590a = 510 b = 810 c = 1150A = 580 B = 1430 C = 250