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LESSON 6: SOLVING PROBLEMS INVOLVING MORE THAN ONE RIGHT TRIANGLE Learning Outcome: Learn to use trigonometry to solve problems modeled by more than one right triangle. Key Math Learnings: Many problems involve one or more right triangles that may be in different planes. The Muttart Conservatory in Edmonton has four climate-controlled square pyramids, each representing a different climatic zone. Each of the tropical and temperate pyramids is 24m high and the side length of its base is 26m. Work with a partner. Use this square based pyramid and label its height and base with the measurements above. Draw right triangles on the drawing that would help you determine the angle between the edges of the pyramid at its apex. How could you use trigonometry to help you determine this angle? A Need to use Pythagorean theorem to solve 24m E B C 13m D π΄πΆ = β24² + 13² = 27.29 πΌπ βπ΄πΆπΈ: tan π΄ = 13 13 = π‘ππβ1 ( ) = 25.46Λ 27.29 27.29 πΌπ βπ΄π·πΈ: β π·π΄πΈ = 2(25.46Λ) = 51Λ We can use trigonometry to solve problems that can be modeled using right triangles. When more than one right triangle is involved, we have to decide which triangles to start with. Ex. Calculate the length of CD to the nearest tenth of a centimetre. C B 47Λ 26Λ A 4.2cm D We need to use βπ΄π΅π· to calculate the length of BD. Angle of elevation and angle of depression: Both angles of elevation and angles of depression are always measured from the horizontal. The angle of elevation looks from the horizontal upwards: Angle of elevation And the angle of depression looks from the horizontal downwards: Angle of depression For example: In Triangle ABC, the angle of elevation and the angle of depression are indicated. Angle of Depression Angle of Elevation Note: The angle of elevation and the angle of depression are always equal Ex. From a height of 50m in his fire tower, a ranger observes the beginnings of two fires. One fire is due west at an angle of depression of 9Λ. The other fire is due east at an angle of depression of 7Λ. What is the distance between the two fires? 9Λ 7Λ 50 m Need to solve for each triangle separately, and recognize that the angle of elevation and depression are equal. 7Λ 9Λ 9Λ 7Λ Ex. A surveyor stands at a window on the 9th floor of an office tower. He uses a clinometers to measure the angles of elevation and depression of the top and the base of a taller building. The surveyor sketches this plan of his measurements. Determine the height of the taller building to the nearest tenth of a metre. 31Λ 42Λ 39m Assignment: pg. 18-121 #1-9, 11, 13-14, 18, 20