Trigonometry Cosine Rule By Mr Porter A B C b a c Q P R x 78°15’ m m 1 angle and 3 sides

You only need to learn ‘1’ of the formulae from each. Definition The cosine rule can be applied to any triangle. For any triangle ABC: to find a side, use : to find an angle, use : A B C b a c These formulae are CYCLIC!

b a c 1) Label the triangle’s sides a, b, and c. 2) Write down the cosine rule for this triangle, a as subject. Example 1: Find the length of side BC (correct to 3 sig. fig.) A B C 8 cm 12 cm 52°30’ 3) Substitute values for A, b, and c. Use your calculator to evaluate the RHS. [Think about it! a < 12+8 (?) [Like Pythagoras’ Thm, take √ Example 2: Find the length of side x (correct to 3 sig. fig.) Q P R x 78°15’ m m r q p 1) Label the triangle’s sides q, p, and r. 2) Write down the cosine rule for this triangle, q as subject. 3) Substitute values for Q, q, p and r. Use your calculator to evaluate the RHS. [Think about it! x < (?)

B Example 3: Find the size of angle ABC (correct to the nearest minute) A C 12 cm 21 cm 18 cm b a c θ 1) Label the triangle’s sides a, b, c and θ. 2) Write down the angle cosine rule for this triangle, B as subject. 3) Substitute values for B, a, b and c. Use your calculator to evaluate the RHS. Therefore, angle ABC is 86° 25’. Example 4: Find the size of angle θ (correct to the nearest minute) 20 cm Q P 42 cm 54 cm R θ p q r 1) Label the triangle’s sides q, p, and r. 2) Write down the angle cosine rule for this triangle, Q as subject. 3) Substitute values for Q, p, q and r. Use your calculator to evaluate the RHS. [Think: Angle -> cos -1 (…) ] Therefore, angle θ = 116°35’.

θ Example 5: A soccer goal is 8 m wide. A player shoots for goal (along the ground) when 18 m from one post and 12 m from the other post. Within what angle (correct to the nearest minute) must the shot be made for the player to have a chance of scoring a goal? 8 m 12 m 18 m 1 angle and 3 sides  cosine rule!! 1) Label the triangle’s sides A, B, C, a, b, and c. A B C a b c 2) Write down the cosine rule for this triangle, c as subject. 3) Substitute values for a, b and c. Use your calculator to evaluate the RHS. [Think: Angle -> cos -1 (…) ] The angle is 20° 45’ Example 6: A boat at sea spots two light houses. Light House A is 8 km away, on a bearing 350°T. Light House B is 5 km away, on a bearing 250°T. Find the distance between the two Light Houses (correct to 3 sig. fig.). Light A Light B C - Boat 350° 255° 5 km 8 km Draw a diagram! Find any angles by simple geometry. 350° – 250° = 100° 100° 1) Label the triangle’s sides a, b, c and C. a b c 2) Write down the side cosine rule for this triangle, c as subject. 3) Substitute values for a, b and C. Use your calculator to evaluate the RHS. The distance is 10.1 km.