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Trigonometry

Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. ?. Trigonometry.

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Trigonometry

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  1. 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 anglesand 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.

  2. Trigonometry Trigonometry is concerned with the relationship between the anglesand 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

  3. Trigonometry Trigonometry is concerned with the relationship between the anglesand 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

  4. Trigonometry Trigonometry is concerned with the relationship between the anglesand 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

  5. Trigonometry Trigonometry is concerned with the relationship between the anglesand 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. ? What’s he going to do next? 45o

  6. Trigonometry Trigonometry is concerned with the relationship between the anglesand 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. ? What’s he going to do next? 45o 324 m

  7. Trigonometry Trigonometry is concerned with the relationship between the anglesand 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. 45o 324 m 324 m

  8. Trigonometry 324 m • 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.

  9. The Trigonometric Ratios A hypotenuse adjacent B C opposite C opposite B adjacent hypotenuse A S O H C A H T O A Make up a Mnemonic!

  10. The Trigonometric Ratios (Finding an unknown side). Example 1. In triangle ABC find side CB. A T C C S T C S S T O O A O O A O A O A H H A H H H H A Diagrams not to scale. 12 cm 70o C B Opp Example 2. In triangle PQR find side PQ. P 22o Q R 7.2 cm Example 3. In triangle LMN find side MN. L 4.3 m M 75o N

  11. 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 C T O O A H H A True Values (2 dp) Sin 30o = 0.50 Cos 30o = 0.87 Tan 30o = 0.58 43.5 m xo 30o 75 m The Trigonometric Ratios (Finding an unknown angle).

  12. Example 1. In triangle ABC find angle A. A T C S S T C C T S O A O O O A O A O A A H A H H H H H Key Sequence Key Sequence Key Sequence 12 cm Tan-1(4.3  1.2) = Sin-1(11.3  12) = Cos-1(7.2  7.8) = C B 11.3 cm Example 2. In triangle LMN find angle N. L 4.3 m M Diagrams not to scale. 1.2 m N Example 3. In triangle PQR find angle Q. P 7.8 cm Q R 7.2 cm The Trigonometric Ratios (Finding an unknown angle).

  13. 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. 15 miles H B 6.4 miles L SOH CAH TOA

  14. 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. 12 ft 9.5 ft Lo SOH CAH TOA

  15. Applications of Trigonometry P 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 ofW. It then returns to base. (a) Make a sketch of the flight. (b) Find the bearing of Q from P. Not to Scale 570 miles 430 miles Q W SOH CAH TOA

  16. Angles of Elevation and Depression. An angle of elevation is the angle measured upwardsfrom 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 25o Angle of elevation 25o Horizontal Explain why the angles of elevation and depression are always equal.

  17. 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. 20 m P 45 m SOH CAH TOA

  18. D C 55 m D 100 m Or more directly since the angles of elevation and depression are equal. 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. SOH CAH TOA

  19. 30o 60o 57 m 30o x m Or more directly since the angles of elevation and depression are equal. 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? SOH CAH TOA

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