SOS HGS Grade 12, 2016

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♦️Trigonometry- Solution of Triangles♦️ Sine rule: Sides of a triangle are proportional to the sine of the angles opposite to them. So, in ΔABC, sin A/a = sin B/b = sin C/c = 2Δ/abc. This may also be written as (a/sin A) = (b/sin B) = (c/sin C) Cosine rule: In any ΔABC, cos A = (b2 + c2 – a2) /2bc cos B = (a2 + c2 – b2)/2ac cos C = (a2 + b2 - c2)/2ab Trigonometric ratios of half-angles: sin A/2 = √[(s-b)(s-c)/bc] sin B/2 = √[(s-c) (s-a)/ac] sin C/2 = √[(s-a) (s-b)/ab] cos A/2 = √s(s - a)/bc cos B/2 = √s(s - b)/ac cos C/2 = √s(s - c)/ab tan A/2 = √[(s - b) (s - c)/s(s - a)] tan B/2 = √[(s - c) (s - a)/s(s - b)] tan C/2 = √[(s - a) (s - b)/s(s - c)] Projection rule: In any ΔABC, a = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A Area of a triangle If Δ denotes the area of the triangle ABC, then it can be calculated in any of the following forms: Δ = 1/2 bc sin A = 1/2 ca sin B = 1/2 ab sin C Δ = √s(s - a)(s – b)(s - c) Δ = 1/2. (a2 sin B sin C)/ sin(B + C) = 1/2. (b2 sin C sin A)/ sin (C + A) = 1/2. (c2 sin A sin B)/ sin (A + B) Semi-perimeter of the triangle If S denotes the perimeter of the triangle ABC, then s = (a + b + c)/2 Napier’s analogy In any ΔABC, tan [(B – C)/2] = (b – c)/(b + c) cot A/2 tan [(C – A) /2] = (c – a)/(c + a) cot B/2 tan [(A – B) /2] = (a – b)/(a + b) cot C/2 m-n theorem Consider a triangle ABC where D is a point on side BC such that it divides the side BC in the ratio m: n, then as shown in the figure, the following results hold good: Triangle ABC(m + n) cot θ = m cot α – n cot ß. (m + n) cot θ = n cot B – m cot C. Apollonius theorem In a triangle ABC, if AD is the median through A, then AB2 + AC2 = 2(AD2 + BD2). If the three sides say a, b and c of a triangle are given, then angle A is obtained with the help of the formula tan A/2 = √(s - b) (s - c) / s(s - a) or cos A = b2 + c2 - a2 / 2bc. Angles B and C can also be obtained in the same way. If two sides b and c and the included angle A are given, then tan (B – C) /2 = (b – c)/ (b + c) cot A/2 This gives the value of (B- C)/2. Hence, using (B + C)/2 = 90o - A/2 along with the last equation both B and C can be evaluated. Now, the sides can be evaluated using the formula a = b sin A/sin B or a2 = b2 + c2 – 2bc cosA. If two sides b and c and the angle B (opposite to side b) are given, then using the following results, we can easily obtain the remaining elements sin C = c/b sinB, A = 180o – (B + C) and b = b sin A/sinB Don't forget join and share this group 👇 👇Join and share ▮@Qubee_Aplus_Education 💡 ▮@Qubee_Aplus_Education 💡 SABA  GUDDAA WAAN TAANEEF MIIDIYAA GUDDAATU NUUF MALA! 📡MIIDIYAA📡 @GAADDISA_SABAA1 Kitaaba Barataa @Qubee_Aplus_Education 🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟 💧 https://t.me/+h4yscp_Pwyg3Mjdk 💧 https://t.me/+h4yscp_Pwyg3Mjdk 🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟

♦️Trigonometry- Solution of Triangles♦️ Sine rule: Sides of a triangle are proportional to the sine of the angles opposite to them. So, in ΔABC, sin A/a = sin B/b = sin C/c = 2Δ/abc. This may also be written as (a/sin A) = (b/sin B) = (c/sin C) Cosine rule: In any ΔABC, cos A = (b2 + c2 – a2) /2bc cos B = (a2 + c2 – b2)/2ac cos C = (a2 + b2 - c2)/2ab Trigonometric ratios of half-angles: sin A/2 = √[(s-b)(s-c)/bc] sin B/2 = √[(s-c) (s-a)/ac] sin C/2 = √[(s-a) (s-b)/ab] cos A/2 = √s(s - a)/bc cos B/2 = √s(s - b)/ac cos C/2 = √s(s - c)/ab tan A/2 = √[(s - b) (s - c)/s(s - a)] tan B/2 = √[(s - c) (s - a)/s(s - b)] tan C/2 = √[(s - a) (s - b)/s(s - c)] Projection rule: In any ΔABC, a = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A Area of a triangle If Δ denotes the area of the triangle ABC, then it can be calculated in any of the following forms: Δ = 1/2 bc sin A = 1/2 ca sin B = 1/2 ab sin C Δ = √s(s - a)(s – b)(s - c) Δ = 1/2. (a2 sin B sin C)/ sin(B + C) = 1/2. (b2 sin C sin A)/ sin (C + A) = 1/2. (c2 sin A sin B)/ sin (A + B) Semi-perimeter of the triangle If S denotes the perimeter of the triangle ABC, then s = (a + b + c)/2 Napier’s analogy In any ΔABC, tan [(B – C)/2] = (b – c)/(b + c) cot A/2 tan [(C – A) /2] = (c – a)/(c + a) cot B/2 tan [(A – B) /2] = (a – b)/(a + b) cot C/2 m-n theorem Consider a triangle ABC where D is a point on side BC such that it divides the side BC in the ratio m: n, then as shown in the figure, the following results hold good: Triangle ABC(m + n) cot θ = m cot α – n cot ß. (m + n) cot θ = n cot B – m cot C. Apollonius theorem In a triangle ABC, if AD is the median through A, then AB2 + AC2 = 2(AD2 + BD2). If the three sides say a, b and c of a triangle are given, then angle A is obtained with the help of the formula tan A/2 = √(s - b) (s - c) / s(s - a) or cos A = b2 + c2 - a2 / 2bc. Angles B and C can also be obtained in the same way. If two sides b and c and the included angle A are given, then tan (B – C) /2 = (b – c)/ (b + c) cot A/2 This gives the value of (B- C)/2. Hence, using (B + C)/2 = 90o - A/2 along with the last equation both B and C can be evaluated. Now, the sides can be evaluated using the formula a = b sin A/sin B or a2 = b2 + c2 – 2bc cosA. If two sides b and c and the angle B (opposite to side b) are given, then using the following results, we can easily obtain the remaining elements sin C = c/b sinB, A = 180o – (B + C) and b = b sin A/sinB 👇Join and share ▮@Qubee_Aplus_Education 💡 ▮@Qubee_Aplus_Education 💡

🌟🌟PHYSICS FORMULAS🌟🌟 1. Area = Length × Breadth 2. Volume = Length × Breadth × Height 3. Mass Density Density= Mass/I 4. Frequency Frequency =1/period 5. Velocity Velocity = Displacement / Time 6. Speed ​​= distance / time 7. Acceleration Acceleration = Velocity / Time 8. Force = Mass × Acceleration 9. Impulse Impulse = Force × Time 10. Work Work = Force × Distance 11. Energy Energy = Force × Distance 12. Power Power = work / time 13. Momentum = Mass × Velocity 14. Pressure = Force / Area 15. Stress Stress = Force / Area 16. Strain Strain = Change in Dimension / Original Dimension 17. Coefficient of elasticity= stress/strain 18. Surface tension = force / length 19. Surface energy = energy / area 20. Velocity gradient = Velocity/distance 21. Pressure gradient 22. Viscosity Coefficient Coefficient of viscosity= Force/(Area × Velocity gradient) 23. Angle Angel = arc/radius 24. Trigonometric ratio Trigonometric ratio = length / length Angular velocity = angle / time 26. Angular Acceleration = Angular Velocity / Time 27. Angular momentum = moment of inertia × angular velocity 28. Moment of inertia = mass × (radius of revolution) 2 29. Torque = Force × Distance 30. Angular frequency = 2π × frequency Universal constant of gravity = force × (distance)2/(mass)2 32. Planck's constant Plank's constant = energy / frequency 33. Specific heat = thermal energy / (mass × temperature) 34. Heat capacity = heat energy 35. Boltzmann's constant = energy / heat 36. Stefan's constant = (energy/area × time)/(heat)4 37. Gas constant Gas constant = (pressure × volume)/(mole × temperature) 38. Charge = current × time 39. Potential difference = Work/Charge 40. Resistance Resistance = potential difference / electric current 41. Capacity = charge/potential difference 42. Electric field = Electric force / charge 43. Magnetic field = force / (electric current × length) 44. Magnetic flux = magnetic field × length 45. Inductance Inductance = Magnetic flux / Electric current 46. ​​Wein's constant = wavelength × temperature 47. Conductivity = 1/Resistance 48. Entropy = Thermal Energy / Temperature 49. Latent heat = thermal energy / mass 50. Coefficient of thermal expansion = change in dimension / (original dimension × temperature) nbsp51. Bulk modulus of elasticity = (volume × change in pressure)/change in volume 52. Electric resistivity Electric resistance = (Resistance × Area)/ Length 53. Electric dipole moment = Force moment / Electric field 54. Magnetic dipole moment = Force moment / Magnetic field 55. Magnetic field strength = Magnetic moment / Volume 56. Refractive index = speed of light in vacuum / speed of light in medium 57. Wave number Wave number= 2π / wavelength 58. Radiant power = energy emitted / time 59. Radiant intensity = Radiant power / solid angle Share waliif qoodaa plz🙏🙏 dabalataaf 👇👇👇👇 🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟 💧 https://t.me/+h4yscp_Pwyg3Mjdk 💧 https://t.me/+h4yscp_Pwyg3Mjdk 🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟🌟