1. The Cosine Rule. AB C a b c a 2 =b2b2 +c2c2 -2bccosA o

2. Proving The Cosine Rule. Consider this triangle: AB C a b c We are looking for a formula for the length of side “a”. Start by drawing an altitude CD of length “h”. h D Let the distance from A to D equal “x”. x The distance from D to B must be “c – x”. c-x To find the Cosine Rule we are going to concentrate on the triangle “CDB”. c-x B C a h D

3. b AB C a b c h D x B C a h D Apply Pythagoras to triangle CDB. a 2 =h2h2 +(c - x) 2 Square out the bracket.a 2 =h2h2 +c2c2 -2cx+ x 2 What does h 2 and x 2 make? b2b2 a 2 =b2b2 +c2c2 -2cx What does the cosine of A o equal? cos A o = x Make x the subject: x =bcosA o Substitute into the formula: a 2 =b2b2 +c2c2 -2cbcosA o We now have: a 2 =b2b2 +c2c2 -2bccosA o The Cosine Rule.

4. When To Use The Cosine Rule. The Cosine Rule can be used to find a third side of a triangle if you have the other two sides and the angle between them. All the triangles below are suitable for use with the Cosine Rule: 6 10 65 o L89 o 13.8 6.2 W 147 o 8 11 M Note the pattern of sides and angle.

5. Using The Cosine Rule. Example 1. Find the unknown side in the triangle below: L 5m 12m 43 o Identify sides a,b,c and angle A o a =Lb =5 c =12A o =43 o Write down the Cosine Rule. a 2 =b2b2 +c2c2 -2bccosA o Substitute values and find a 2. a 2 =5252 +12 2 - 2 x 5 x 12 cos 43 o a 2 =25 + 144-(120 x0.731 ) a 2 =81.28Square root to find “a”. a = 9.02m

6. Example 2.137 o 17.5 m 12.2 m M Find the length of side M. Identify the sides and angle. a = Mb = 12.2C = 17.5A o = 137 o Write down Cosine Rule and substitute values. a 2 =b2b2 +c2c2 -2bccosA o a 2 = 12.2 2 + 17.5 2 – ( 2 x 12.2 x 17.5 x cos 137 o ) a 2 = 148.84 + 306.25 – ( 427 x – 0.731 ) Notice the two negative signs. a 2 = 455.09 + 312.137 a 2 = 767.227 a = 27.7m

7. What Goes In The Box ? 1. Find the length of the unknown side in the triangles below: (1) 78 o 43cm 31cm L (2) 8m 5.2m 38 o M (3) 110 o 6.3cm 8.7cm G L = 47.5cm M =5.05m G = 12.4cm

8. Finding Angles Using The Cosine Rule. Consider the Cosine Rule again:a 2 =b2b2 +c2c2 -2bccosA o We are going to change the subject of the formula to cos A o Turn the formula around: b 2 + c 2 – 2bc cos A o = a 2 Take b 2 and c 2 across.-2bc cos A o = a 2 – b 2 – c 2 Divide by – 2 bc. Divide top and bottom by -1 You now have a formula for finding an angle if you know all three sides of the triangle.

9. Finding An Angle. Use the formula for Cos A o to calculate the unknown angle x o below: xoxo 16cm 9cm11cm Write down the formula for cos A o Identify A o and a, b and c. A o = x o a = 11b = 9c = 16 Substitute values into the formula. Calculate cos A o. Cos A o =0.75 Use cos -1 0.75 to find A o A o = 41.4 o Example 1

10. Example 2. Find the unknown angle in the triangle below: 26cm 15cm 13cm yoyo Write down the formula. Identify the sides and angle. A o = y o a = 26b = 15c = 13 Substitute into the formula. Find the value of cosA o cosA o =- 0.723 The negative tells you the angle is obtuse. A o =136.3 o

11. What Goes In The Box ? 2 Calculate the unknown angles in the triangles below: (1) 10m 7m 5m aoao bobo (2) 12.7cm 7.9cm 8.3cm (3) coco 27cm 14cm 16cm a o =111.8 o b o = 37.3 o c o =128.2 o