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Law of Sines

Law of Sines. Information. The law of sines. Here is a non-right angled triangle, △ ABC. C. a is the length of the side opposite angle A. b. a. b is the length of the side opposite angle B. c is the length of the side opposite angle C. A. B. c. The law of sines:. a. b. c.

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Law of Sines

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  1. Law of Sines

  2. Information

  3. The law of sines Here is a non-right angled triangle,△ABC. C a is the length of the side opposite angle A. b a b is the length ofthe side opposite angle B. c is the length ofthe side opposite angle C. A B c The law of sines: a b c sin A sin B sin C = = or = = sin A sin B sin C a b c

  4. Proving the law of sines Prove the law of sines by drawing a perpendicular line segment from any side to the opposite vertex. C h h sinB = sinA = by definition: b a rearrange for h: h = bsinA h = asinB b a h h bsinA = asinB equating: A B c D a b divide both sides of the equation by sinA∙ sinB: = sinA sinB if we had chosen the perpendicular line segment from A to a, we would have found that: c b sinB sinC = a c b = = therefore:  sinA sinB sinC

  5. When to use the law of sines When should we use the law of sines? The law of sines can be used if: C • two angles and the length of a side opposite one of the angles is known b a B A c • or if the length of two sides and the angle opposite one of these sides is known. C b a If we do not have this information, we must use a different method or a different law. B A c

  6. Finding lengths Use the law of sines to find the length of side a. a c b = write the law of sines: = sinA sinB sinC a 7 substitute for the given values: = sin39° sin118° multiply each side of the equation by sin 118°: 7sin118° a = C sin39° evaluate: a= 9.82in (to the nearest hundredth) a 7cm 118° 39° A B

  7. Lengths practice

  8. Finding angles Use the law of sines to find m∠B. a b c = write the law of sines: = sinA sinB sinC sinB sin46° substitute for the given values: = 6 8 multiply each side of the equation by 8: 8sin46° sinB = C 6 8sin46° find the inverse for each side of the equation: = sin–1 m∠B 6 6in 8in evaluate: m∠B = 73.56° (to the nearest hundredth) 46° B A

  9. The ambiguous case

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