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Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. ? Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. 30o Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. 35o Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. 40o Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. ? 45o What’s he going to do next? Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. ? 45o What’s he going to do next? 324 m Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. 324 m 45o 324 m Trigonometry Eiffel Tower Facts: •Designed by Gustave Eiffel. •Completed in 1889 to celebrate the centenary of the French Revolution. •Intended to have been dismantled after the 1900 Paris Expo. •Took 26 months to build. •The structure is very light and only weighs 7 300 tonnes. •18 000 pieces, 2½ million rivets. •1665 steps. •Some tricky equations had to be solved for its design. 1 H 2 H 2 x f (x ) cons tantx (H x ) x xw (x )f (x )dx 324 m The Trigonometric Ratios A adjacent C B hypotenuse Sine A B opposite opposite hypotenuse C Opposite Hypotenuse SinA O H Adjacent Hypotenuse CosA A H Opposite Adjacent TanA O A Cosine A Tangent A adjacent Make up a Mnemonic! A S O H C A H T O A The Trigonometric Ratios (Finding an unknown side). Example 1. In triangle ABC find side CB. S O H C A H T O A A CB Diagrams Sin 700 70o 12 cm 12 not to scale. 12Sin 700 CB 11.3 cm (1dp ) C B Opp Example 2. In triangle PQR find side PQ. S O H C A H T O A P 7.2 7.2 Cos 220 PQ PQ Cos 220 22o Q PQ 7.8 cm (1dp ) R 7.2 cm Example 3. In triangle LMN find side MN. S O H C A H T O A L 4.3 m 4.3 4.3 M MN Tan 750 Tan 750 MN 75o MN 1.2 m (1dp ) N The Trigonometric Ratios (Finding an unknown angle). True Values (2 dp) Sin 30o = 0.50 Cos 30o = 0.87 Tan 30o = 0.58 Anytime we come across a right-angled triangle containing 2 given sides we can calculate the ratio of the sides then look up (or calculate) the angle that corresponds to this ratio. S O H C A H T O A Tanx 0 xoo 30 75 m 43.5 0.58 75 43.5 m The Trigonometric Ratios (Finding an unknown angle). Example 1. In triangle ABC find angle A. S O H C A H T O A A 12 cm C 11.3 cm 11.3 Sin A 12 Key Sequence Sin-1(11.3 12) = 0 Angle A 70 (nearest deg ree ) B Example 2. In triangle LMN find angle N. S O H C A H T O A L 4.3 m Key Sequence M 4.3 Tan N Tan-1(4.3 1.2) = 1.2 1.2 m Diagrams not o Angle N 7 4 (nearest degree) N to scale. Example 3. In triangle PQR find angle Q. S O H C A H T O A P 7.8 cm Key Sequence 7.2 Cos Q -1(7.2 7.8) = Cos 7.8 Q R 7.2 cm Angle Q 23o (nearest degree) Applications of Trigonometry A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the trip and calculate the bearing of the harbour from the lighthouse to the nearest degree. H 15 miles B 15 Tan L 6.4 Angle L 66.90 6.4 miles Bearing 360 66.9 293o L SOH CAH TOA Applications of Trigonometry A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. Calculate the angle that the foot of ladder makes with the ground. 9.5 Sin L 12 o Angle L 52 12 ft 9.5 ft Lo SOH CAH TOA Applications of Trigonometry An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 430 miles North to a point P before turning left and flying for 570 miles to a second point Q, West of W. It then returns to base. Not to Scale P (a) Make a sketch of the flight. (b) Find the bearing of Q from P. 570 miles 430 Cos P 570 430 miles Angle P 41o Bearing 180 41 221 0 Q W SOH CAH TOA Angles of Elevation and Depression. An angle of elevation is the angle measured upwards from a horizontal to a fixed point. The angle of depression is the angle measured downwards from a horizontal to a fixed point. Horizontal Angle of depression Explain why the angles of elevation and depression are always equal. 25o Angle of elevation Horizontal 25o Applications of Trigonometry A man stands at a point P, 45 m from the base of a building that is 20 m high. Find the angle of elevation of the top of the building from the man. Tan P 20 45 Angle P 240 (nearest deg ree ) 20 m 45 m P SOH CAH TOA A 25 m tall lighthouse sits on a cliff top, 30 m above sea level. A fishing boat is seen 100m from the base of the cliff, (vertically below the lighthouse). Find the angle of depression from the top of the lighthouse to the boat. 100 Tan C 55 Angle C 61.2o Angle D 90 61.20 290 (nearest deg ree ) C D 55 m 100 m D Or more directly since the angles of elevation and depression are equal. SOH CAH TOA Tan D 55 Angle D 29o 100 A 22 m tall lighthouse sits on a cliff top, 35 m above sea level. The angle of depression of a fishing boat is measured from the top of the lighthouse as 30o. How far is the fishing boat from the base of the cliff? x Tan 60 57 x 57Tan 60 =99m (nearest m) 30o 60o 57 m 30o xm SOH CAH TOA Or more directly since the angles of elevation and depression are equal. Tan 30 57 x x 57 99m Tan 30