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Mathematics for Australia Year 9 - Homework - 2013/1/4 15:04 - page 90 Mathematics for Australia Year 9 - Homework TRIGONOMETRY (Chapter 17) CHAPTER 17: TRIGONOMETRY 17A b Hence estimate the value of: i sin 34± ii cos 34± LABELLING RIGHT ANGLED TRIANGLES iii tan 34± REMINDER c Check your answers using a calculator. hypotenuse opposite µ adjacent d Explain the difference between your results in b and c. 1 For each diagram below, name the: i hypotenuse ii side opposite angle µ iii side adjacent to angle µ. a A b µ B C P c X µ d Z Q 2 µ R L µ M a Use the diagram alongside to determine which is greater: i sin 55± or sin 70± 55° ii tan 55± or tan 70± 70° Y N 17B b Check your answers using a calculator. THE TRIGONOMETRIC RATIOS 3 Consider the triangle PQR alongside. a Use Pythagorasâ€™ theorem to find the unknown side length. REMINDER OPP . sin µ = HYP ADJ cos µ = . HYP OPP tan µ = . ADJ 1 Consider the right angled triangle ABC alongside. a Use a ruler to find the length of each side, to the nearest millimetre. HYP P 15.3 mm R 23° 36.0 mm Q Give your answer correct to 1 decimal place. OPP µ ADJ A 34° B b Hence, estimate the value of: ii cos 23± i sin 23± C 91 0 iii tan 23± Mathematics for Australia Year 9 - Homework - 2013/1/4 15:04 - page 91 TRIGONOMETRY (Chapter 17) 17C Mathematics for Australia Year 9 - Homework 19° f FINDING SIDE LENGTHS x cm REMINDER Step 1: On the figure mark HYP, OPP, and ADJ relative to a given angle. Step 2: Choose an appropriate trigonometric ratio, and construct an equation. Step 3: 18.5 cm 3 Find, rounded to one decimal place, all unknown sides and angles: Solve the equation to find the unknown side length. 25° 4m bm am µ 1 Write down a trigonometric equation connecting the angle and the sides given: a b x 72° y c 30° b c 17D r 55° FINDING ANGLES q REMINDER 2 Find x, giving your answer rounded to 2 decimal places: a 128 cm 73° x cm xm b Step 1: On the figure mark HYP, OPP, and ADJ relative to the angle you are trying to find. Step 2: Choose an appropriate trigonometric ratio, and construct an equation. (The LHS will be sin µ, cos µ, or tan µ.) Step 3: Use inverse sine, inverse cosine or inverse tangent on your calculator to find the value of µ. 47° 27 m 1 Find, to one decimal place, the measure of the angle marked µ in: a 10 cm µ c x mm 62° d 33° 9 cm 93 mm 35 mm 50 mm b 45 mm x mm µ e c 3.2 m 40° 4.5 m xm 91 1 3.6 m µ Mathematics for Australia Year 9 - Homework - 2013/1/4 15:04 - page 92 Mathematics for Australia Year 9 - Homework d TRIGONOMETRY (Chapter 17) In this section, round your answers to 3 significant figures. 2.1 km µ 1 A telescope is set 1.2 m up to monitor the movement of Venus 26° across the night sky. eye level The telescope is at an angle of 26± to the horizontal, and the telescope is 1:2 m long. How many centimetres above eye level is the other end of the telescope? 0.8 km 2 44.7 mm a Find, rounded to one decimal place, all unknown sides and angles in the following: ¯ ® xm 46.6 mm 2 A sandwich board is set up on the footpath. The angle at the apex is 32± , and the feet are set 40 cm apart. How long are the boards that make up the sign? b Check your answer for x using Pythagorasâ€™ theorem. 3 Try to find µ in the following using trigonometry. What conclusions can you draw? µ H A E S E 32° 8.96 m 40 cm 8.96 m 9m 3 Find the unknown angles in this trapezium. ¯ ¯ 6m ® 17E ® 15 m PROBLEM SOLVING WITH TRIGONOMETRY REMINDER Use these steps to solve problems involving right angled triangles: Step 1: Step 2: 4 A triangular block of land is being measured by a surveyor. What are the values of the unknown angles? Draw a diagram to illustrate the situation. Mark on the diagram the unknown angle or side that needs to be calculated. We often use x for a length and µ for an angle. Step 3: Locate a right angled triangle in your diagram. Step 4: Write an equation connecting an angle and two sides of the triangle using an appropriate trigonometric ratio. Step 5: Solve the equation to find the unknown. Step 6: Write your answer in sentence form. 91 2 30 m 18 m Á µ Mathematics for Australia Year 9 - Homework - 2013/1/4 15:04 - page 93 TRIGONOMETRY (Chapter 17) Mathematics for Australia Year 9 - Homework 5 Find the perimeter of the rhombus shown alongside. 3° 6 An aeroplane, currently at altitude 1500 m, starts its final descent into the airport. Its angle of descent is 3± . In kilometres, how far away from the airport runway is the plane when it begins its descent? mm 25 20° 1500 m REVIEW OF CHAPTER 17 1 Find the exact value of: a sin µ b cos µ 8 4 µ c tan µ 5.5 km 3 km 18 m 12° 2 Write a trigonometric equation connecting the angle and the sides given: 3 Find the value of ®: a ® 105 cm 7 A ten pin bowler releases the ball from the right of his lane. It travels in a straight line to hit the 7-pin as shown. What is the angle µ between the path of the ball and the side of the lane? µ a b 8 Find the lengths of the diagonals of this rectangle. b 4.55 m 7.6 m ® 10.8 m P S 34° Q R 9 Determine the height of a boy who casts a 3 metre shadow when the sun is 20± above the horizon. 4 Find the value of x: a x cm 49° b 21 cm 11 mm 10 a Find the exact value of x in the triangle given. ® 42 cm x mm ¯ 2 cm 8° 5 Find the remaining side lengths in the given triangle, giving your answer correct to the nearest millimetre: x cm A b Hence, or otherwise, find the unknown angles of the triangle. 35° 1m C B 91 3