Sine and Cosine rules

1. Sine and Cosine rules Applied to non-right angled triangles

2. hyp opp A adj Introduction • In Sec 2,you have learnt to apply the trigonometric ratios to right angled triangles.

3. How can trigonometry be applied to triangles which are not right angled?

4. B • Let’s refer to the triangles below: • In DABC • The side opposite angle A is called a. • The side opposite angle B is called b. c a C A b P • In DPQR • The side opposite angle P is called p. • And so on q r R Q p

5. There are two new rules known as the • sine rule and • cosine rule.

6. C b a A D c Dividing by sinA and sinB gives: In the same way: 1. The sine rule B Using DBDC: DC = a sinB. Draw the perpendicular from C to meet AB at D. Using DADC: DC = b sinA. Therefore asinB = bsinA. Putting both results together: The proof needs some changes to deal with obtuse angles.

7. B Example 1 a 5.3 cm 85o • Find the length of BC. C 32o A b • Substitute A = 32o, C = 85o, b = 5.3: • Multiply by sin32o:

8. X (or , impossible here). 110o Example 2 17 cm y Z • Find the size of angle Z. Y 20 cm • Substitute: X= 110o, x = 20, z = 17: leading to • Multiply by 17: leading to

9. Points to note when using the sine rule to find an angle: • 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 since sin x = sin (180-x).

10. 2. The cosine rule B c a C A b

11. h D c – x x c Press to skip proof Proof of the cosine rule C b a Applying Pythagoras’ Theorem to ADC gives: h2 = b2 – x2 Applying Pythagoras’ Theorem to DBC gives: a2 = h2 + (c – x)2 = h2 + c2 – 2cx + x2. k Substituting from equation  into equation k gives: a2 = b2 – x2 + c2 – 2cx + x2 = b2 + c2 – 2cx. l Using DAC again: x = bcosA . Substituting this into l gives: a2 = b2 + c2 – 2cb cosA . i.e. A B a2 = b2 + c2 – 2bccosA Again the proof needs some changes to deal with obtuse angles.

12. Cosine rule can be used in two • one for finding a side, • one for finding an angle.

13. The cosine rule for finding a side. B c a • 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. C A b

14. Starting from: The cosine rule for finding an angle. Add 2bc cosA and subtract a2 getting 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. B c Divide both sides by 2bc: a C A b R q p Q P r

15. Substitute B = 79o, a = 8.4, c = 11.2 into B Example 3 11.2 cm 8.4 cm 79o A C • Find the length of AC. b (Show what you are calculating, but you do not need to show intermediate results.)

16. D Example 4 7 cm 5 cm M R • Substitute d = 10, r = 5, m = 7 10 cm • Find the size of angle D. into There is no need to show intermediate results. getting leading to The cosine rule automatically takes care of obtuse angles.

17. 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 know either two sides and the included angle or all three sides.