The Cosine Rule. A B C a b c a2 = b2 + c2 -2bccosAo.

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Presentation transcript:

The Cosine Rule. A B C a b c a2 = b2 + c2 -2bccosAo

A B C a b c h D x c-x c-x B C a h D Apply Pythagoras to triangle CDB. a2 = h2 + (c - x) 2 a2 = h2 + c2 -2cx + x2 Square out the bracket. a2 = b2 + c2 -2cx b2 What does h2 and x2 make? a2 = b2 + c2 -2cbcosAo What does the cosine of Ao equal? We now have: x cos Ao = Make x the subject: b a2 = b2 + c2 -2bccosAo Substitute into the formula: x = bcosAo The Cosine Rule.

When To Use The Cosine Rule. The Cosine Rule can be used to find a third side of a triangle if you have the other two sides and the angle between them. All the triangles below are suitable for use with the Cosine Rule: 6 10 65o L 89o 13.8 6.2 W 147o 8 11 M Note the pattern of sides and angle.

Using The Cosine Rule. Example 1. Find the unknown side in the triangle below: L 5m 12m 43o Identify sides a,b,c and angle Ao Write down the Cosine Rule. c = 12 Ao = 43o a = L b = 5 a2 = b2 + c2 -2bccosAo Substitute values and find a2. a2 = 52 + 122 - 2 x 5 x 12 cos 43o a2 = 25 + 144 - (120 x 0.731 ) a2 = 81.28 Square root to find “a”. a = 9.02m

Example 2. Find the length of side M. 137o 17.5 m 12.2 m M Identify the sides and angle. a = M b = 12.2 C = 17.5 Ao = 137o Write down Cosine Rule and substitute values. a2 = b2 + c2 -2bccosAo a2 = 12.22 + 17.52 – ( 2 x 12.2 x 17.5 x cos 137o ) a2 = 148.84 + 306.25 – ( 427 x – 0.731 ) Notice the two negative signs. a2 = 455.09 + 312.137 a2 = 767.227 a = 27.7m

What Goes In The Box ? 1. Find the length of the unknown side in the triangles below: (1) 78o 43cm 31cm L G = 12.4cm (3) 110o 6.3cm 8.7cm G L = 47.5cm (2) 8m 5.2m 38o M M =5.05m

Finding Angles Using The Cosine Rule. Consider the Cosine Rule again: a2 = b2 + c2 -2bccosAo We are going to change the subject of the formula to cos Ao Turn the formula around: b2 + c2 – 2bc cos Ao = a2 -2bc cos Ao = a2 – b2 – c2 Take b2 and c2 across. Divide by – 2 bc. Divide top and bottom by -1 You now have a formula for finding an angle if you know all three sides of the triangle.

Finding An Angle. Example 1 Use the formula for Cos Ao to calculate the unknown angle xo below: xo 16cm 9cm 11cm Ao = xo a = 11 b = 9 c = 16 Write down the formula for cos Ao Identify Ao and a , b and c. Substitute values into the formula. Cos Ao = 0.75 Calculate cos Ao . Ao = 41.4o Use cos-1 0.75 to find Ao

Example 2. Find the unknown angle in the triangle below: 26cm 15cm 13cm yo Write down the formula. Identify the sides and angle. Substitute into the formula. Ao = yo a = 26 b = 15 c = 13 Find the value of cosAo The negative tells you the angle is obtuse. cosAo = - 0.723 Ao = 136.3o

What Goes In The Box ? 2 Calculate the unknown angles in the triangles below: (1) 10m 7m 5m ao (3) co 27cm 14cm 16cm ao =111.8o (2) 12.7cm 7.9cm 8.3cm bo bo = 37.3o co =128.2o