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7.2 – The Sine Ratio A. Sine Ratio Introduction: Another relationship exists between the ratio of the length of the side opposite a given angle to the length of the hypotenuse. In two similar triangles this ratio will always be the same. It is known as the __________________________. sin length of side opposite length of hypotenuse sin Example 1: Use your scientific calculator to determine the following sine ratios. Round to four decimal places. Note: make sure your calculator is in degree mode. a) sin 10º c) sin 50º b) sin 45º d) sin 70º Example 2: Calculate the angle to the nearest degree. a) sin A = 0.5164 Math 10 Workplace b) sin B = 0.8461 Marsh B. Labelling Triangles In order to solve angles and side in right triangles, it is important to be able to label them properly. The three sides are: ______________________, ______________________, ______________________, The _________________ is always directly opposite the right angle and it is the longest side. The ______________ side is always next to the angle of interest. The _______________ side is always directly across the angle of interest. A Example 3: Consider ∆ABC θ Label all of the sides given θ and ϕ. ϕ B C Worksheet: Introduction to Trigonometry Introduction to Trigonometry Worksheet 1) Calculate the following sine ratios. Round to four decimal places. a) sin 15 ° c) sin 80 ° b) sin 60° d) sin 100 ° 2) Calculate the value the angle to the nearest degree. a) sin 0.5879 c) sin b) sin Math 10 Workplace 0.9994 d) sin 0.2635 0.4569 Marsh 3) For each triangle below, name: (i) the hypotenuse (ii) the side opposite the angle marked θ (iii) the side adjacent the angle marked θ. a) b) c) d) e) f) g) h) i) j) Answers: 1) a) 0.2588 b) 0.8660 c) 0.9848 d) 0.9848 2) a) 36° b) 88° c)15° d) 27° Math 10 Workplace Marsh A C. Three types of Trigonometry Questions: 1) Solving for the angle Solve for A: sin A 11.2 ft 6.4 ft opp hyp Steps: B 7.3 ft C 1) Label triangle according to the angle of interest (in this case A) – hyp, opp, and adj. 2) Write down required formula (1 mark). 3) Fill in known values and leave unknown value (1 mark). 4) Solve. For the angle you must use the inverse function of sine. (1 mark) Practice: a) b) Math 10 Workplace Marsh A 2) Solving for the top of the ratio sin C opp hyp 4.3 cm c 46º Steps: B C 1) Label triangle according to the angle of interest (in this case C) – hyp, opp, and adj. 2) Write down required formula (1 mark). 3) Fill in known values and leave unknown value (1 mark). 4) Solve. (1 mark) Practice: a) b) Math 10 Workplace Marsh 3) Solving for the bottom of the ratio sin C opp hyp Steps: 1) Label triangle according to the angle of interest (in this case C) – hyp, opp, and adj. 2) Write down required formula (1 mark). 3) Fill in known values and leave unknown value (1 mark). 4) Solve. (1 mark) Practice: a) b) Worksheet: The Sine Ratio Math 10 Workplace Marsh The Sine Ratio Worksheet 1) Use your calculator to determine the value of each of the following sine ratios to four decimal places. a) sin 30° b) sin 48° c) sin 62° d) sin 77° 2) Calculate the angle to the nearest degree. a) sin D = 0.5491 b) sin H = 0.9998 3) Solve for the indicated angle in the following diagrams. a) b) c) Math 10 Workplace Marsh d) e) 4) Find the opposite side in the following diagrams. Round answers to one decimal place. a) b) c) d) Math 10 Workplace Marsh 5) Find the length of the hypotenuse, to one decimal place, in the following diagrams. a) b) c) d) 6) A rafter makes an angle of 28° with the horizontal. If the rafter is 15 feet long, what is the height at the rafter’s peak? Draw a diagram. Math 10 Workplace Marsh 7) How high is a weather balloon that is tied to the ground if it is attached to a 15 metre string and the angle between the string and the ground is 35°? Draw a diagram. 8) How long is a guy wire that is attached 4.2 metres up a pole if it makes an angle of 52° with the ground? Draw a diagram. 9) A boat is carried with the current at an angle of 43° to the shore. If the river is approximately 15 metres wide, how far does the boat travel before reaching the opposite shore? Draw a diagram. Answers: 1) a) 0.5000 b) 0.7431 c) 0.8829 d) 0.9744 2) a) 33° b) 89° 3) a) 38.7° b) 39.8° c) 43.4° d) 32.4° e) 50° 4) a) 10.8 cm b) 11.3 cm c) 2.8 mm d) 8.7 km 5) a) 199.8 mm b) 8.6 m c)154.7 mm d) 25.6 mm 6) 7.04 ft 7) 8.6m 8) 5.3 m 9) 22m Math 10 Workplace Marsh D. Angle of Elevation and Depression Trigonometry is useful in many real life situations. Two common types of angles are the angles of elevation and depression. When you look up the _____________ _____________________ is the angle formed between the horizontal and your line of sight. When you look down the ___________ ____________________ is the angle formed between the horizontal and your line of sight. Example: You are flying a kite overhead. The angle of elevation is 65º. The length of string used is 75 ft. How high is the kite? Angle of Elevation and Depression Worksheet 1) George is in a hot air balloon that is 125 metres high. The angle of elevation from a house below, to the balloon, is 18°. How far is George from the house? 2) The angle of elevation of a road is 4.5°. What is the length of the section of road if it rises 16 metres? 3) The angle of elevation of a slide that is 3.6 metres long is 32°. How high above the ground is the top of the slide? Math 10 Workplace Marsh 4) A ramp with a length of 21.2 metres has an angle of elevation of 15°. How high up does it reach? 5) The angle of elevation from the bottom of a waterslide to the platform above is 20°. If the waterslide is 25 metres long, how high is the platform? 6) A man walks at an angle of 68° north of east for 45 metres. How north of his starting point is he? Answers: 1) about 404.5 m 2) about 203.9 m 5) about 8.6 m 6) about 41.7 m 3) about 1.9 m 4) about 5.5 m Math 10 Workplace Marsh