# Using the Cosine rule

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Information about Using the Cosine rule
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Published on October 3, 2008

Author: iphillips

Source: authorstream.com

The Cosine Rule : The Cosine Rule The sine and cosine rules are used to solve sides and angles in NON right-angled triangles to find the third side when you know two sides and the inclusive angle The cosine rule has two main uses these are :- to find an angle when you know the length of all the sides Labelling of triangle : Labelling of triangle It is usually a good idea to take the information from the problem given and transfer it to your own diagram. Label the angles Use Capitol letters C B A Label the sides Use lower case letters c a b Note sides are opposite the angles Using the cosine rule : Using the cosine rule a2 = b2 + c2 – 2bc cosA C B A c a b There are three forms of the cosine rule c b A We use this form if given Using the cosine rule : Using the cosine rule b2 = a2 + c2 – 2ac cosB C B A c a b There are three forms of the cosine rule c a B We use this form if given Using the cosine rule : Using the cosine rule c2 = a2 + b2 – 2ab cosC C B A c a b There are three forms of the cosine rule a b C We use this form if given Worked example 1 : Worked example 1 Find the unknown side C B A c a b 200 130 250 First Label the diagram Worked example 1 : Worked example 1 Find the unknown side C B A c a b 200 130 250 Choose the appropriate cosine rule b2 = a2 + c2 – 2ac cosB Worked example 1 : Worked example 1 Find the unknown side C B A c a b 200 130 250 b2 = 1302 + 2002 – 2x130x200 cos25 Substitute values Worked example 1 : Worked example 1 Find the unknown side C B A c a b 200 130 250 Substitute values b2 = 56900 – 47128 99 b = 99 Worked example 2 : Worked example 2 Find the unknown side C B A c a b 220 135 1200 Choose the appropriate cosine rule c2 = a2 + b2 – 2ab cosC Worked example 2 : Worked example 2 Find the unknown side C B A c a b 220 135 1200 c2 = 2202 + 1352 – 2x220x135 cos120 Worked example 2 : Worked example 2 Find the unknown side C B A c a b 220 135 1200 c2 = 2202 + 1352 – 2x220x135 cos120 Two negative multiplied gives positive c2 = 66625 + 29700 c = 96325 Worked example 2 : Worked example 2 Find the unknown side C B A c a b 220 135 1200 c = 310 310 Worked example 3 : Worked example 3 Find angle A C B A c a b 220 135 320 Choose the appropriate cosine rule 2bc cosA b2 + c2 a2 – = b2 + c2 a2 – = cosA 2bc Worked example 3 : Worked example 3 Find angle A C B A c a b 220 135 320 Substitute values b2 + c2 a2 – = cosA 2bc Worked example 3 : Worked example 3 Find angle A C B A c a b 220 135 320 Check you get this answer using your calculator You now need to use the inverse cosine function shown as cos-1 on the calculator You will usually have to press the shift key first, followed by the COS key You should find that the angle is 33.3o 33.3o Summary of the Cosine Rule : Summary of the Cosine Rule find the third side when you know two sides and the inclusive angle The cosine rule can be used to :- find an angle when you know the length of all the sides

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