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Sine Rule & Cosine Rule These are two extremely useful trignometric results which are applicable to all triangles, not just right angled ones. To prove the Sine Rule, consider three identical copies of the same triangle with sides a,b,c and (opposite) angles A,B,C. Divide each into two right angled triangles. b C b y x b C A A z a a a c c B B x = b sin C z = a sin C z = c sin A y = a sin B x = b sin A x = c sin B ∴ b sin C = c sin B ∴ a sin B = b sin A sin C sin B = c b sin B sin A = b a Hence the Sine Rule states c sin A sin B sin C = = a b c or ∴ a sin C = c sin A sin C sin A = c a a b c = = sin A sin B sin C Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 1 To prove the Cosine Rule, consider three identical copies of the same triangle with sides a,b,c and (opposite) angles A,B,C. Divide each into two right angled triangles, as per the Sine Rule derivation. However, in this case we will combine some basic trigonometry with Pythagoras’ theorem to write two expressions for the dividing line x, y or z. b b y b C A x a z a a c c c B b 2 = ( b cos A ) + y 2 c 2 = ( c cos B ) + x 2 a 2 = ( a cos C ) + z 2 a 2 = ( c − b cos A ) + y 2 b 2 = ( a − c cos B ) + x 2 c 2 = ( b − a cos C ) + z 2 2 2 2 2 ∴ b 2 − ( b cos A ) = a 2 − ( c − b cos A ) 2 2 2 2 ∴ c 2 − ( c cos B ) = b 2 − ( a − c cos B ) 2 2 ∴ a 2 − ( a cos C ) = c 2 − ( b − a cos C ) 2 2 c 2 − c 2 cos 2 B = b 2 − {a 2 − 2ac cos B + c 2 cos 2 B} a 2 − a 2 cos 2 C = c 2 − {b 2 − 2ab cos C + a 2 cos 2 C} b 2 − b 2 cos 2 A = a 2 − c 2 + 2bc cos A − b 2 cos 2 A c 2 − c 2 cos 2 B = b 2 − a 2 + 2ac cos B − c 2 cos 2 B a 2 − a 2 cos 2 C = c 2 − b 2 + 2ab cos B − a 2 cos 2 C a 2 = b 2 + c 2 − 2bc cos A b 2 = c 2 + a 2 − 2ac cos B c 2 = a 2 + b 2 − 2ab cos C b2 + c2 − a 2 A = cos 2bc c2 + a 2 − b2 B = cos −1 2ac a2 + b2 − c2 C = cos −1 2ab b 2 − b 2 cos 2 A = a 2 − {c 2 − 2bc cos A + b 2 cos 2 A} −1 Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 2 Examples: Use the cosine rule to find angles given three other sides For finding angles it is best to use the Cosine Rule, as cosine is single valued in the range 0o... 180o whereas sine has two values. If the angle is obtuse (i.e. > 90o), then the sine rule can yield an incorrect answer since most calculators will only give the solution to sinθ = k within the range -90o .... 90o sin θ = 1 2 cos θ = 1 2 θ = 30o ,150o ,... θ = 60o ,300o ,... TWO values in the range 0o... 180o Which one is correct? Only one solution in the range 0o... 180o cosθ sinθ is the x coordinate of the unit circle is the y coordinate of the unit circle θ is measured anti-clockwise from the x axis Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 3 Surveying example: Use the measurements below to work out the height H of Mount Everest above sea level Surveying example (2): Note the calculation can also be done more generally, and directly, using tan 180o − 24o − 135o = 21o x H 135o 45o 24o 10,000m The plains of Nepal 823m H α = 45o y h = 823m Sea level Sine rule: x 10, 000 = o sin 24 sin 21o 10, 000 sin 24o ∴x = sin 21o H − 823 = x sin 45o = x 2 x 2 10, 000sin 24o ∴ H = 823 + sin 21o × 2 ∴ H = 8848m ∴ H = 823 + The method of determining altitude of mountain peaks using elevation measurements at either end of a flat baseline was used to great effect in the Great Trigonometrical Survey of India which continued for much of the 19th century. George Everest was the second superintendent of the GTS, and the world’s highest peak was (renamed) after him. In Nepal it is known as Sagarmatha, and in Tibet, Chomolungma “Mother Goddess of the Universe”. β = 24o x = 10,000m The plains of Nepal Sea level H − h = ( x + y ) tan β H − h = y tan α H −h ∴y = tan α H −h ∴ H − h = x tan β + tan β tan α tan β tan β H 1 − = h + x tan β − h tan α tan α H ( tan α − tan β ) = h tan α + x tan α tan β − h tan β H = h+ x tan α tan β tan α − tan β 10, 000 tan 45o tan 24o ∴ H = 823 + tan 45o − tan 24o ∴ H = 8848m George Everest (1790-1866) Mathematics topic handout: Trigonometry – Sine rule and Cosine Rule Dr Andrew French. www.eclecticon.info PAGE 4