*Sine and Cosine rulesTrigonometry applied totriangles without right angles. D R Martin

*IntroductionYou have learnt to apply trigonometry to right angled triangles.

*Now we extend our trigonometry so that we can deal with triangles which are not right angled.

*First we introduce the following notation.We use capital letters for the angles, and lower case letters for the sides.In DABCThe side opposite angle A is called a.The side opposite angle B is called b.In DPQRThe side opposite angle P is called p.And so on

*There are two new rules.

*1. The sine ruleDraw the perpendicular from C to meet AB at P.

Using DBPC: PC = a sinB.BUsing DAPC: PC = b sinA.Therefore a sinB = b sinA.Putting both results together:The proof needs some changes to deal with obtuse angles.

*Example 1Find the length of BC.Substitute A = 35o, B = 95o, b = 6.2: Multiply by sin35o:

*Example 2Find the size of angle R.Substitute: P = 100o, p = 18, r = 15: Multiply by 15:leading toleading to

*Try to avoid finding the largest angle as you probably do not know whether you want the acute or the obtuse angle. The largest angle is opposite the longest side, the smallest angle opposite the shortest side. When there are two possible angles they add up to 180 degrees.When using the sine rule to find an angle:

*Now do: Page 106: questions 1 a and b, 2 a and b, and 3. Note misprints: in 1a top line should be 6 cm long (not 7 cm) in 2b right side should be 7cm (not 6 cm). Check your answers.

*2. The cosine rule

*Proof of the cosine ruleApplying Pythagoras Theorem to DAPC gives: h2 = b2 x2 jApplying Pythagoras Theorem to DBPC gives:a2 = h2 + (c x)2 = h2 + c2 2cx + x2. kSubstituting from equation j into equation k gives:a2 = b2 x2 + c2 2cx + x2 = b2 + c2 2cx. lUsing DAPC again: x = bcosA .Substituting this into l gives: a2 = b2 + c2 2cb cosA .i.e.a2 = b2 + c2 2bc cosA Again the proof needs some changes to deal with obtuse angles.Press to skip proof

*There are two main ways of writing the cosine ruleone for finding a side,one for finding an angle.

*The formula starts and ends with the same letter, one lower case, one capital.The square of a side is the sum of the squares of the other 2 sides minus twice the product of the 2 known sides and the cosine of the angle between them.The cosine rule for finding a side.

*The cosine of an angle of a triangle is the sum of the squares of the sides forming the angle minus the square of the side opposite the angle all divided by twice the product of first two sides.The cosine rule for finding an angle.Add 2bc cosA and subtract a2 gettingDivide both sides by 2bc:

*Example 3Find the length of BC.(You need to show what you are calculating, but you do not need to show intermediate results.)

*Example 4Find the size of angle D.Substitute d = 8, r = 4, m = 6The cosine rule automatically takes care of obtuse angles.intogettingleading to There is no need to show intermediate results.

*Now do: Page 109: questions 1 a and c,2 a and c,and 3. Check your answers.

*How do I know whether to use the sine rule or the cosine rule?To use the sine rule you need to know an angle and the side opposite it.You can use it to find a side (opposite a second known angle) or an angle (opposite a second known side).To use the cosine rule you need to knoweither two sides and the included angleor all three sides.

*Now do: Page 111: questions 1 and 5.Page 115: questions 5 and 6. Check your answers.