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The Sine Rule

The Sine Rule. Sine and Cosine rules. Trigonometry applied to triangles without right angles. hyp. opp. A. adj. Introduction. You 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. B.

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The Sine Rule

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  1. The Sine Rule

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

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

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

  5. B P q c a r R Q C p A b • First we introduce the following notation. • We use capital letters for the angles, and lower case letters for the sides. • In DABC • The side opposite angle A is called a. • The side opposite angle B is called b. • In DPQR • The side opposite angle P is called p. • And so on

  6. C b a A P c The sine rule Draw the perpendicular from C to meet AB at P. B 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.

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

  8. 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

  9. G B 3. 1. 2. F 53º 13 cm 41º x 8.0 35.3 5.5 x 62º A x 130º 28º D E 5 cm 63º 76º C H 26 mm I 10.7 4. 5.2 cm 5. 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

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

  11. 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 = x sin43º × x = 10.33 cm

  12. sinA a sinB b sinC c = = 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:

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

  14. 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 = sin xº 7× sin xº = 0.850 x = sin-1 0.850 = 58.3º

  15. 7.6 cm 1. 2. 3. 82º 105º 6.5cm 47º 5 cm 8.2 cm 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 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

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

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