Presentation on theme: "Mr Barton’s Maths Notes"— Presentation transcript:
1Mr Barton’s Maths Notes Trigonometry4. Sine and Cosine Rules
24. 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 RulesYou 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!
3C 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 sideWhat 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!ExampleCabBAcxMultiply both sides by sin 37
4C 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!ExampleCabBAcxMultiply both sides by 16
5a2 = b2 + c2 – 2bcCosA a2 = b2 + c2 – 2bcCosA C b a B A c 3. The Cosine Rule – Finding an unknown SideWhat 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!ExampleCaba2 = b2 + c2 – 2bcCosABAca2 = b2 + c2 – 2bcCosAx2 = – 2 x 5.2 x 4.5 x Cos58x2 = – 2 x 5.2 x 4.5 x Cos58xx2 = …Square root both sidesx = 4.74m (2dp)
6C 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!ExampleCabBAcx
7Finding Sides Finding Angles Cosine Rule Sine Rule Cosine Rule 17 ? 11 A Nice Little SummaryCosine Rule17?118a2 = b2 + c2 – 2bcCosA?65o1410Sine Rule??91062o55o43o16Finding SidesFinding AnglesCosine RuleSine RuleNeed 2 sides and included angleNeed all 3 sidesNeed 2 angles and any sideNeed 2 sides and an angle not included