Mr Barton’s Maths Notes Trigonometry 4. Sine and Cosine Rules www.mrbartonmaths.com

4. Sine and Cosine Rules The Big Problem with Trigonometry As far as mathematical things go, Pythagoras, and the trio of Sin, Cos and Tan, were pretty good… weren’t they? However, they had one major draw back… They only worked for right-angled triangles! That certainly limited their use. Well, imagine if we had some rules which worked for… wait for it… any triangle! Well, you’ll never guess what… we do!... The Sine and Cosine Rules! The Crucial Point about the Sine and Cosine Rules You must know when to use each rule… what information do you need to be given? If you can get your head around that, then it’s just plugging numbers into formulas! Note: In all the formulas, small letters represent sides, and Capital Letters represent Angles!

C b a B A c x 1. The Sine Rule – Finding an unknown Side What Information do you need to be given? Two angles and the length of a side What is the Formula? Remember: If you are given two angles, you can easily work out the 3rd by remembering that angles in a triangle add up to 1800! Example C a b B A c x Multiply both sides by sin 37

C b a B A c x 2. The Sine Rule – Finding an unknown Angle What Information do you need to be given? Two lengths of sides and the angle NOT INCLUDED (i.e. not between those two sides!) What is the Formula? Remember: If the angle is included, you will have to use the Cosine Rule! Example C a b B A c x Multiply both sides by 16

a2 = b2 + c2 – 2bcCosA a2 = b2 + c2 – 2bcCosA C b a B A c 3. The Cosine Rule – Finding an unknown Side What Information do you need to be given? Two sides of the triangle and the INCLUDED ANGLE (i.e. the angle between the two sides!) What is the Formula? Remember: You must be pretty good on your calculator to get these ones correct! Example C a b a2 = b2 + c2 – 2bcCosA B A c a2 = b2 + c2 – 2bcCosA x2 = 5.22 + 4.52 – 2 x 5.2 x 4.5 x Cos58 x2 = 5.22 + 4.52 – 2 x 5.2 x 4.5 x Cos58 x x2 = 22.48977… Square root both sides x = 4.74m (2dp)

C b a B A c x 4. The Cosine Rule – Finding an unknown Angle What Information do you need to be given? All three lengths of the triangle must be given! What is the Formula? Remember: This is just a re-arrangement of the previous formula, so you only need to remember one! Example C a b B A c x

Finding Sides Finding Angles Cosine Rule Sine Rule Cosine Rule 17 ? 11 A Nice Little Summary Cosine Rule 17 ? 11 8 a2 = b2 + c2 – 2bcCosA ? 65o 14 10 Sine Rule ? ? 9 10 62o 55o 43o 16 Finding Sides Finding Angles Cosine Rule Sine Rule Need 2 sides and included angle Need all 3 sides Need 2 angles and any side Need 2 sides and an angle not included

Good luck with your revision!