# The Sine Rule C. McMinn. SOH/CAH/TOA can only be used for right-angled triangles. The Sine Rule can be used for any triangle: AB C a b c The sides are.

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The Sine Rule C. McMinn

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

Example 1: C B A 76 º 7cm Find the length of BC x a sinA c sinC b c a = x sin76º 7 sin63 º =× sin76ºsin76º × x = 7 sin63º × sin76º x = 7.6 cm 63 º Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

Example 2: Q R P 55 º 82 º 15cm Find the length of PR x p sinP q sinQ rq p = 15 sin82º x sin43 º =× sin43ºsin43º × = x 15 sin82º sin43º × x = 10.33 cm 43 º Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use.

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

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

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! sinA a sinB b sinC c= Alternative form of the Sine Rule:

Example 1: A B C 72 º 6cm Find the size of angle ABC x º sinA a sinB b b a c = sin72º 6 sin xº 4 =× 44 ×4 × = sin xº 4 ×4 × sin xº = 0.634 Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. 4cm sin72º 6 x = sin -1 0.634 = 39.3º

Example 2: R Q P 85 º 8.2cm Find the size of angle PRQ x º sinP p sinR r q r p = sin85º 8.2 sin xº 7 =× 77 ×7 × = sin xº 7 ×7 × sin xº = 0.850 7cm sin85º 8.2 x = sin -1 0.850 = 58.3º

1. 2.3. 4. 5. 6. 7. 47 º 6 cm xºxº 5 cm xºxº 105 º 8.8 cm 6.5cm xºxº 33 º 5.2 cm 5.5 cm xºxº 7.6 cm 8.2 cm xºxº 82 º 8 m 70 º 9.5 m (Be careful!) xºxº 27 º 6 km 3.5 km 74 º xºxº 7 mm 9 mm 37.6 66.6 45.5 31.0 51.1 57.7 92.1 52.3 º 22.9 º

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

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