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The Cosine Rule.

C. b. a. A. B. c. a 2 =. b 2. +. c 2. -2bccosA o. The Cosine Rule. C. b. a. h. h. A. B. x. c. D. C. a. c-x. B. D. Proving The Cosine Rule. Consider this triangle:. We are looking for a formula for the length of side “a”. c-x.

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The Cosine Rule.

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

  2. C b a h h A B x c D C a c-x B D Proving The Cosine Rule. Consider this triangle: We are looking for a formula for the length of side “a”. c-x Start by drawing an altitude CD of length “h”. To find the Cosine Rule we are going to concentrate on the triangle “CDB”. Let the distance from A to D equal “x”. The distance from D to B must be “c – x”.

  3. C b a c-x h h A B x c D C a a2 = b2 + c2 -2bccosAo c-x B 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 Substitute into the formula: x = bcosAo The Cosine Rule.

  4. 10 W 65o 6 6.2 L 89o 13.8 11 8 147o M 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: Note the pattern of sides and angle.

  5. L 5m 43o 12m a2 = b2 + c2 -2bccosAo Using The Cosine Rule. Example 1. Find the unknown side in the triangle below: Identify sides a,b,c and angle Ao Write down the Cosine Rule. c = 12 Ao = 43o a = L b = 5 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

  6. 17.5 m 137o 12.2 m M a2 = b2 + c2 -2bccosAo Example 2. Find the length of side M. Identify the sides and angle. a = M b = 12.2 C = 17.5 Ao = 137o Write down Cosine Rule and substitute values. 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

  7. 43cm (1) 78o 31cm L 6.3cm (3) 110o G (2) 8.7cm M 5.2m 38o 8m What Goes In The Box ? 1. Find the length of the unknown side in the triangles below: G = 12.4cm L = 47.5cm M =5.05m

  8. a2 = b2 + c2 -2bccosAo Finding Angles Using The Cosine Rule. Consider the Cosine Rule again: 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.

  9. 9cm 11cm xo 16cm Example 1 Finding An Angle. Use the formula for Cos Ao to calculate the unknown angle xo below: 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

  10. yo 13cm 15cm 26cm Example 2. Find the unknown angle in the triangle below: 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

  11. (1) 7m ao 5m (3) 14cm 10m 27cm co 12.7cm 16cm (2) 8.3cm 7.9cm What Goes In The Box ? 2 Calculate the unknown angles in the triangles below: ao =111.8o bo bo = 37.3o co =128.2o

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