Topical Revision Worksheet 8 Trigonometry EM Solution (KK)

Topical Revision Worksheet 8 Topic: Trigonometry EM Question 1 [2008 EOY, Q17] marks The diagram below shows the position of three locations A, B and C with respect to a reference point O. The bearings of A, B and C from O are , and respectively. It is also given that m and m. (i) Find the bearing of B from A. Solution: Hence bearing of B from A [3] (ii) If the bearing of B from C is , calculate the length of OC. Solution: and Hence by sine rule m [3] (iii) A mast stands vertically at B such that the angle of elevation of the top of the mast from O is . Find the height of the mast. Solution: height m [2] A B C O N

Question 2 [2009 EOY, Q14] marks The diagram below shows three circular fields with centres P, Q and R tangential to one another. The radii of the three fields are 2 m, 4 m and x m respectively and ∠PRQ = 60°. (i) Form an equation in x and show that it reduces to . Solution: Using cosine rule, [3] (ii) Solve the equation, giving your answer in surd form and find the exact area of triangle PQR. Solution: Area = m 2 [4] P Q R x 2 4 60°

(i) A pole of y m stands at P. A boy walks from R to Q. Given that the maximum angle of elevation of the top of the pole during the walk is 60°, find y. Solution: Maximum angle of elevation occurs when distance, d from P to QR is the smallest. [3] Question 3 [2010 EOY, Q9] marks Three points A, B and C on level ground are such that A is due north of B, km, the bearing of C from B is and km. (i) Find the possible distances of A from B. Solution: By sine rule, [M1] Hence and So and [5] P y 60° d

(ii) Taking the shorter distance between A and B, determine the bearing of A from C. Solution: Bearing of from C [2] Question 4 [2011 EOY, Q7] marks The variables x and y are related by the equation . A straight line passes through (5, 7) is obtained when is plotted against . (i) Given that the gradient of this line is 2, calculate the value of a and of b. Solution: A1 [3] (ii) Given that this line also passes through (k, 13), find the value of k. Solution: [2] Question 5 [2011 EOY, Q15] marks In the diagram A, B and S represent three points in the sea. B is 3 km from A on a bearing of The bearings of S from A and B are and respectively. Calculate the distance AS. Solution: [3]

(ii) A ship at S sails directly to A at a steady speed of 2 . After 40 minutes, the ship is at T, a point on AS. (a) Calculate the distance BT. Solution: Distance = Speed x Time km AT = 4.226 – 1.333 km = 2.893 km….M1 Using cosine rule, [3] (b) Find the bearing of B from T. Solution: The bearing of B from T is [3] (iii) A hot air balloon, H, is lowering at a point 600 m vertically above B. Calculate the greatest angle of elevation of H from an observer on the ship as the ship sails from S to A. Solution: Shortest distance from observer to B = [2]

Question 6 [2012 EOY Q13] marks ABC represents a horizontal triangular field such that A is due north of B, AB = 240 m, AC = 180m and the bearing of C from A is 110°. (i) Find the distance BC. Solution: Using Cosine Rule [3] (ii) Find the bearing of C from B. Solution: Using Sine Rule Bearing of C from B is 043.5°. [3] (iii) A vertical tree, of height 25 m, is planted at C. A farmer walks along a footpath from A to B. Find the maximum angle of elevation of the top of the tree from the farmer during this walk. You may ignore the height of the farmer. Solution: Maximum angle of elevation occurs when distance between tree and farmer is the shortest. Let this shortest distance be d. Let the maximum angle of elevation be θ [4]

Question 7 [2013 EOY, Q12] marks In the diagram, PQR represents a horizontal triangular field such that PR = 85 m and PQ = 40 m. The bearing of R from P and the bearing of Q from P is and respectively. (i) Calculate the distance from Q to R. Solution: In [2] (ii) Calculate the area of the triangular field PQR. Solution: Area of the triangular field [2] (iii) Calculate the shortest distance from P to the road QR. Solution: In Hence the shortest distance from P to QR is [2] TP represents a building at the corner P. Given that the angle of depression of the top of the building T to the corner R is , (iv) calculate the height of the building TP. Solution: In , Hence [2] A man walks along the road from Q to R, (v) calculate the greatest angle of elevation of the top of the building. Solution: The biggest angle of elevation is [2]

Question 8 [2014 EOY, Q14] marks Three points, S, Y and T are on level ground. Y is the foot of a vertical flagpole XY, and S is 30 m due south of Y. (i) The angle of elevation of X from S is . Calculate the height of the flagpole. Solution: Height of the flagpole XY [2] (ii) The bearing of T from S and the bearing of T from Y are respectively, calculate the distance ST. Solution: The bearing of T from S and the bearing of T from Y are respectively. [3] (iii) A man is standing at the midpoint of ST. Calculate the angle of elevation of X from the man. Solution: Let the midpoint of S and T be M. Using Cosine Rule, Angle of elevation of X from the man [5] T N X S 30 Y

Question 9 [2015 EOY, Q11] marks A man, at point M, is 80 metres at a bearing of from an observation deck, D, which stands 25 metres tall. He begins to walk to a point P which is due East of D at a bearing of . Ignoring the man’s height, calculate (i) the angle of elevation of the top of the deck from the man at M, Solution: Angle of elevation [2] (ii) the distance he walks when the angle of elevation of the top of the deck from his position is the greatest, and Solution: Distance he walks m or 69.3 (3 s.f.) m [3] (iii) the distance DP. Solution: By Sine Rule, [2]