4.1 Reviewing the Trigonometry of Right Triangles

Transcription

1 4.1 Reviewing the Trigonometry of Right Triangles INVSTIGT & INQUIR In the short story The Musgrave Ritual, Sherlock Holmes found the solution to a mystery at a certain point. To find the point, he had to start near the stump of an elm tree and take 20 paces north, then 10 paces east, then 4 paces south, and finally 2 paces west. 1. Let be the point where Holmes started pacing and S be the point where he stopped. raw a diagram of his path. Join S. raw another line segment so that S is the of a right triangle. 2. What are the lengths of the perpendicular sides of the right triangle? 3. What trigonometric ratio can you use to find from the lengths of the perpendicular sides? 4. Find, to the nearest degree. 5. What methods could you use to find the length of S, in paces? 6. What is the length of S, to the nearest pace? 7. In what direction, and for how many paces, could Holmes have walked in order to go directly from to S? 8. fter Holmes arrived at S, he looked back at the location of point on the ground. What angle did his line of sight make with the ground, to the nearest degree? ssume that the ground was level and that the height of his eyes above the ground was approximately equal to the length of 2 paces. 266 MHR hapter 4

2 The primary trigonometric ratios are sine = cosine = tangent = adjacent adjacent adjacent To solve a right triangle means to find all the unknown sides and the unknown angles. XMPL 1 Solving a Right Triangle, Given a Side and an ngle In, = 90, = 18.6, and b = 11.3 cm. Solve the triangle by finding a) the unknown angle b) the unknown sides, to the nearest tenth of a centimetre SOLUTION a) Using the given information, = = 71.4 b) From the diagram, a = sin a = 11.3 sin 18.6 = 3.6 c = cos c = 11.3 cos 18.6 = 10.7 In, = 71.4, a = 3.6 cm, and c = 10.7 cm. b =11.3 cm 18.6 c a Using the mode settings, ensure that the calculator is in degree mode. 4.1 Reviewing the Trigonometry of Right Triangles MHR 267

3 XMPL 2 Solving a Right Triangle, Given Two Sides In F, = 90, d = 7.4 m, and f = 6.5 m. Solve the triangle by finding a) the unknown angles, to the nearest tenth of a degree b) the unknown side, to the nearest tenth of a metre SOLUTION a) From the diagram, 7.4 tan = 6.5 F = 48.7 F = = 41.3 b) From the diagram, sin 48.7 = 7.4 e e d = 7.4 m e sin 48.7 = 7.4 e = 7.4 sin 48.7 f = 6.5 m = 9.9 In F, = 48.7, F = 41.3, and e = 9.9 m. If you are standing on a cliff beside a river, and you look down at a boat, the angle that your line of sight makes with the horizontal is called the angle of depression. If you look up at a helicopter, the angle that your line of sight makes with the horizontal is called the angle of elevation. ngle of elevation ngle of depression Horizontal 268 MHR hapter 4

4 XMPL 3 Western Red edars athedral Grove, on Vancouver Island, is a rain forest of firs and western red cedars. From a point 40 m from the foot of one cedar, the angle of elevation of the top is 65. Find the height of the cedar, to the nearest metre. SOLUTION raw and label a diagram. Let h represent the height of the cedar. h = tan h = 40tan 65 = 86 h The cedar is 86 m tall, to the nearest metre m On the arth, a parallel of latitude is a circle parallel to the equator. XMPL 4 Parallel of Latitude Find the length of the 35 parallel of latitude, to the nearest 10 km. ssume that the radius of the arth is 6380 km km 35 parallel SOLUTION In the diagram, is the centre of the arth, and is a point on the equator. is the centre of the circle defined by the 35 parallel, and is a point on its circumference. is the radius, r, of the 35 parallel. is a right angle. =, because both are radii of the arth. = (alternate angles) 6380 km r km 4.1 Reviewing the Trigonometry of Right Triangles MHR 269

5 In, r 6380 = cos 35 r = 6380cos 35 = 5226 The length of the 35 parallel of latitude is its circumference,. = 2πr = 2π(5226) stimate = The length of the 35 parallel of latitude is km, to the nearest 10 km. To copy the previous answer to the cursor location, enter ns by pressing 2nd ( ). XMPL 5 Rock Pillars Rock pillars are interesting geological features found in several national parks in Ontario. Rock pillars, found in rivers and lakes, have been sculpted by wind and water. geologist wanted to determine the height of a rock pillar in a river. The geologist set up a theodolite at and measured to be baseline was marked off, perpendicular to. The length of is 10 m, and = If the height of the theodolite is 1.6 m, what is the height of the rock pillar, to the nearest tenth of a metre? SOLUTION alculate the length of, and then add the height of the theodolite to determine the height of the rock pillar. In, = tan In, = tan 10 = tan = tan 28.5 = 10 tan 56.4 = MHR hapter 4 = 15.1tan 28.5 = = 9.8 So, the height of the rock pillar is 9.8 m, to the nearest tenth of a metre.

6 Key oncepts For any acute angle in a right triangle, adjacent sin = cos = tan = adjacent adjacent To use trigonometry to solve a right triangle, given the measure of one acute angle and the length of one side, find a) the measure of the third angle using the angle sum in the triangle b) the measure of the other two sides using the sine, cosine, or tangent ratios To use trigonometry to solve a right triangle, given the lengths of two sides, find a) the measure of one angle using its sine, cosine, or tangent ratio b) the measure of the third angle using the angle sum in the triangle c) the measure of the third side using a sine, cosine, or tangent ratio To use trigonometry to solve a problem involving two right triangles, a) use a diagram showing the given information and the unknown side length(s) or angle measure(s) b) identify the two triangles that can be used to solve the problem, and plan how to use each triangle c) carry out the plan ommunicate Your Understanding 1. escribe how you would solve, given = 90, = 36, and c = 12 cm. 2. escribe how you would solve RST, given S = 90, s = 22 cm, and t = 15 cm. 3. escribe the difference between an angle of elevation and an angle of depression. 4. escribe how you would find the measure of m 6.5 m 4.1 Reviewing the Trigonometry of Right Triangles MHR 271

7 Practise 1. Solve each triangle. Round each side length to the nearest unit and each angle to the nearest degree. a) b) c = 56 cm 33 a c) U d) t = 10 m S u b s = 15 m T p = 8 cm f = 60 m 41 Q R d F r = 13 cm q e P 3. Solve each triangle. Round answers to the nearest tenth, if necessary. a) In XYZ, X = 90, x = 9.5 cm, z = 4.2 cm b) In KLM, M = 90, K = 37, m = 12.3 cm c) In, = 90, = 55.1, b = 4.8 m d) In F, = 90, d = 18.2 cm, f = 14.9 cm 4. Find the measure of, to the nearest tenth of a degree. a) 2. Solve each triangle. Round each side length to the nearest tenth of a unit and each angle to the nearest tenth of a degree. a) W X y 9.6 cm 63.5 x Y b) 28.8 cm cm 5 cm 7 cm b) L 57.4 m n 20.3 m M 18 cm N c) d) G i K c) R 24.5 m I H 12.8 m 72.3 cm J 68.8 cm j L S m T 27.2 m U 272 MHR hapter 4

8 d) 11.7 m 15.6 m 8. Find RT, to the nearest centimetre. R 38 F Find, to the nearest tenth of a metre. 6. Find RS, to the nearest tenth of a centimetre. P G m 9. Find MN, to the nearest tenth of a centimetre. S 43 cm 52 T N M L 5.8 cm U 22.7 K Q 9.8 cm R S 7. Find FH, to the nearest tenth of a metre m F G 10. Find, to the nearest metre m 8.7 m m pply, Solve, ommunicate H 11. Mica am The highest dam in anada is the Mica am, one of three dams on the olumbia River in ritish olumbia. From a point 600 m from the foot of the dam, the angle of elevation of the top of the dam is 22. What is the height of the dam, to the nearest metre? 12. rctic ircle Find the length of the rctic ircle, which is north, to the nearest 10 km. ssume that the radius of the arth is 6380 km. 4.1 Reviewing the Trigonometry of Right Triangles MHR 273

9 13. Surveying surveyor measured the height of a vertical rock face by determining the measurements shown. If the surveyor s theodolite had a height of 1.7 m, find the height of the rock face,, to the nearest tenth of a metre m ommunication a) Find the area of F, to the nearest tenth of a square metre. b) What other minimum sets of conditions (sides, angles) would allow you to calculate the area of F? 14.8 m Latitude a) Find the length of the 20 parallel of latitude, to the nearest 10 km. ssume that the radius of the arth is 6380 km. b) Find the length of the parallel of latitude where you live, to the nearest 10 km. 16. pplication From 1857 to 1860, Great ritain financed the construction of ten lighthouses in ritish North merica. They were built because obsolete navigational aids were hindering economic growth. The lighthouses are called the Imperial Lights. Four of them were built along the approaches to the Saint Lawrence, and six were built on the eastern shore of Lake Huron. The Point lark lighthouse, on Lake Huron, is 28.3 m tall. From the top of the lighthouse, the angle of depression of a ship is 3.3. How far is the ship from the lighthouse, to the nearest metre? 17. Inquiry/Problem Solving Show that the length of any parallel of latitude is equal to the length of the equator times the cosine of the latitude angle. 18. Measurement Find the volume of the triangular prism, to the nearest cubic centimetre. F 56 cm 274 MHR hapter cm

10 19. Great Pyramid The Great Pyramid of Khufu has a square base with a side length of about 230 m. The four triangular faces of the pyramid are congruent and isosceles. The altitude of each triangular face makes an angle of 52 with the base. Find the measure of each base angle of the triangular faces, to the nearest degree. 20. Geometry a) is an acute triangle. Show that the area,, of can be found using the formula = 0.5acsin. b) Show that the area,, of can also be found using the c formula = 0.5absin. c) Find the other formula, similar to those in parts a) and b), for the area of. d) escribe the pattern in the formulas. e) What given information is sufficient to find the area of an acute triangle? xplain. f) Find the area of the triangle to the right, to the nearest tenth of a square metre m base angle a X b 8.5 m g) Find the area of the triangle to the right, to the nearest tenth of a square centimetre. h) o the formulas from parts a), b), and c) apply to obtuse triangles? xplain and justify your reasoning using diagrams. Y 19.5 cm m 72.5 Z cm F HIVMNT heck Knowledge/Understanding Thinking/Inquiry/Problem Solving ommunication pplication On a wall, a spider is 100 cm above a fly. The fly starts moving horizontally at the speed of 10 cm/s. fter 1 s, the spider begins moving at twice the speed of the fly, in such a way as to intercept the fly by taking a straight line path. In what direction does the spider move, and how far has the fly moved when they meet? 4.1 Reviewing the Trigonometry of Right Triangles MHR 275