The Cosine Law - The Burns Home Page
The Cosine Law - The Burns Home Page
The Cosine Law - The Burns Home Page
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<strong>The</strong> <strong>Cosine</strong> <strong>Law</strong><strong>The</strong> law of <strong>Cosine</strong>s relates the cosine of any angle of a triangle to the lengths of thethree sides of the triangle.Usage of the <strong>Cosine</strong> <strong>Law</strong><strong>The</strong> <strong>Law</strong> of <strong>Cosine</strong>s is used when the following triangle measurements are given:‣ SAS (Side-Angle-Side)‣ SSS (side-Side-side) triangle measurements are given.In the standard notation of triangles, each vertex (which can include the angle) islabelled with a capital letter, and the length of the side opposite that vertex isdenoted by the same letter in lower case.<strong>Cosine</strong> <strong>Law</strong>In any triangle ABC, with lengths a, b, c then: 2 2 2a b c 2 bc cos A2 2 2b a c 2 accosB2 2 2c a b 2 abcosCProof of <strong>Cosine</strong> <strong>Law</strong> : 2 2 2c a b 2 ab cos CAchbBxDayC‣ Draw h perpendicular to BC from A.‣ ABDis a right-triangle‣2 2 2c h x2h ay2h a 2ay y2 2 2 2 2 2h a y 2ay
‣ In ADCyby b cos C‣ cosC‣ ‣ thereforewe haveh y b2 2 22 2 2 2c h y a 2ay 2 2b a 2ab cos C<strong>The</strong> proofs for other relationships are similar.ExampleSolve ABC , if a 9.6 m, b 20.6 m, c 14.7m. Round the side length to the nearesttenth of a metre and the angles to the tenth of a degree.Solution:Since we have a SSS triangle, then the <strong>Cosine</strong> <strong>Law</strong> is used:cosB 2 2 2b a c 2 ac cos B2 2 2b a c2ac20.6 9.6 14.72 9.6 14.72 2 2 0.41138747B 114.3Now apply the Sine <strong>Law</strong> to find AAABsin sina bsinA sin114.39.6 20.6sin 0.4247316241A 25.1To find CC180 114.3 25.140.6
ExampleSolve triangle ABC as shown below10C110°16AcBSolution:Because we have a SAS Triangle problem, we will use the <strong>Cosine</strong> <strong>Law</strong>2 2 2c a b 2 abcosC 2 216 10 2 16 10 cos 110256 100 320cos 110465.44645c 21.6465.44645Let’s again use the <strong>Cosine</strong> <strong>Law</strong> to determine Acos2 2 2a b c 2 bccosA A A2 2 216 10 21.6 2 10 21.6 cos100 466.562564320.7172A 44.2Angle B 180 44.2 110 25.8ExampleTwo roads diverge at a 52 angle. Two bike riders take separate routes at 17km/hand 24km/h. How far apart are they after 2h
Solution:First we must find the distance travelled after two hour2 24 482 17 34<strong>The</strong>refore the sides of the triangle are 34km and 48kmSince we have a contained angle, this is a job for the cosine law.d2 2 234 48 234 48cos52 1156 2304 2009.52...1450.480...d 38<strong>The</strong>refore the two bikers are about 38km apart.
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