The Sine Rule

Trigonometry applied to triangles without right angles. Sine and Cosine rules Trigonometry applied to triangles without right angles.

Introduction You have learnt to apply trigonometry to right angled triangles. A hyp adj opp

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. A a b c C B In DABC The side opposite angle A is called a. The side opposite angle B is called b. Q q p r R P In DPQR The side opposite angle P is called p. And so on

The sine rule Draw the perpendicular from C to meet AB at P. Using DBPC: PC = a sinB. Using DAPC: PC = b sinA. Therefore a sinB = b sinA. Dividing by sinA and sinB gives: In the same way: Putting both results together: The proof needs some changes to deal with obtuse angles.

SOH/CAH/TOA can only be used for right-angled triangles. The Sine Rule can be used for any triangle: C The sides are labelled to match their opposite angles b a A B c a sinA b sinB c sinC = = The Sine Rule:

A Example 1: Find the length of BC 76º c 7cm b 63º C x B a Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. a sinA c sinC = x sin76º 7 sin63º sin76º × = × sin76º 7 sin63º x = × sin76º x = 7.6 cm

G 1. B 3. 2. F 53º 13 cm 41º x 8.0 35.3 5.5 x A 62º x 28º 130º D E 5 cm 63º 76º C H 26 mm I 4. 10.7 5. 5.2 cm x 61º R 6. P 37º 66º 57º 10 m 35º x 5.2 77º 62º Q 12 cm 6 km 85º 7. x 6.6 65º 86º x 6.9

Remember: Draw a diagram Label the sides Set out your working exactly as you have been shown Check your answers regularly and ask for help if you need it

P Example 2: Find the length of PR 82º x r q 43º 55º Q 15cm R p Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. p sinP q sinQ = 15 sin82º x sin43º sin43º × = × sin43º 15 sin82º sin43º × = x x = 10.33 cm

Finding an Angle The Sine Rule can also be used to find an angle, but it is easier to use if the rule is written upside-down! Alternative form of the Sine Rule: sinA a sinB b sinC c = =

C Example 1: Find the size of angle ABC 6cm a 4cm b 72º x º A Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. B c sinA a sinB b = sin72º 6 sin xº 4 4 × = × 4 sin72º 6 4 × = sin xº sin xº = 0.634 x = sin-1 0.634 = 39.3º

P Example 2: Find the size of angle PRQ 85º q 7cm r x º R p 8.2cm Q sinP p sinR r = sin85º 8.2 sin xº 7 7 × = × 7 sin85º 8.2 7 × = sin xº sin xº = 0.850 x = sin-1 0.850 = 58.3º

1. 7.6 cm 2. 3. 47º 82º 105º 6.5cm 5 cm 8.2 cm xº xº xº 8.8 cm 6 cm 5. 66.6° xº 37.6° xº 45.5° xº 8.8 cm 6 cm 5. 6 km 4. 5.5 cm 31.0° xº 27º 3.5 km 51.1° xº 5.2 cm 33º Slide 10 is incomplete. Try to add slides on applications of Sine Rule 7. 6. 8 m 74º 57.7° xº 70º 9 mm 9.5 m 92.1° xº 52.3º (←Be careful!→) 22.9º 7 mm

Remember: Draw a diagram Label the sides Set out your working exactly as you have been shown Check your answers regularly and ask for help if you need it