UNIT 4 MODULE 2: Geometry and Trigonometry

Transcription

1 Year 12 Further Mathematics UNIT 4 MODULE 2: Geometry and Trigonometry CHAPTER 9 - TRIGONOMETRY This module covers the application of geometric and trigonometric knowledge and techniques to various two- dimensional and three- dimensional practical spatial problems. Familiarity with the trigonometric ratios sine, cosine and tangent, similarity and congruence, pythagoras theorem, basic properties of triangles and applications to regular polygons, corresponding, alternate and co- interior angles and angle properties of regular polygons is assumed. Applications, including: Specification of location (distance and direction) in two dimensions using three figure bearings; Calculation of angles and distances in a vertical plane (that is, finding or using angles of elevation and depression); Interpretation and use of a contour map to calculate distances and the average slope between two points; Calculation of unknown angles and distances given triangulation measurements. Question to complete 9A 2, 3, 4, 5, 6, 7 9B 1(a, c, e), 3, 5, 6, 7, 8, 9, 10, 11, 12, 13 9C 1, 2, 3, 4, 5, 6, 9, 10, 11 9D 1, 2, 3, 4, 5, 6, 7, 10, 11 9E 1, 2, 3, 4, 5, 6, 10, 11, 12, 13 9F 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 9G 1, 2, 3, 4, 5, 6, 7 Chapter 9 Page 1 of 26

2 TABLE OF CONTENTS CHAPTER 9 - TRIGONOMETRY A Angles... 3 Using the CAS calculator to convert to degrees, minutes and seconds... 3 Adding and subtracting angles... 5 Using a CAS calculator to add and subtract angles... 5 Some geometry (angle) laws B Angles of Elevation and Depression... 8 Angle of Elevation... 8 Angle of Depression... 8 Angles of elevation and depression are in a vertical plane C Bearings Standard compass bearings Other compass bearings True bearings D Navigation and Specification of Locations Hints Navigation E Triangulation - cosine and sine rules F Triangulation - Similarity G Contour Maps Contour lines and intervals The distance between contour lines indicates the steepness of the slope Map scales Average slope Chapter 9 Page 2 of 26

3 9A Angles An angle can be measured in degrees, minutes and seconds (DMS) 60 minutes = 1 degree 30 minutes = 2 1 or 0.5 degree 15 minutes = 6 minutes = 35 24ʹ is read as 35 degrees 24 minutes Example 1: Convert to degrees and minutes. Using the CAS calculator to convert to degrees, minutes and seconds On a Calculator page, complete the entry line as: (56.75 ) DMS Then press ENTER. Note: the and are obtained by pressing / k and you can just type DMS. Note: The DMS function can also be accessed from the catalogue. k Chapter 9 Page 3 of 26

4 Example 2: Convert ʹ to its decimal form. On a Calculator page, select the degrees, minutes, seconds template from the maths expression templates. This can be accessed by pressing t. Complete the entry line as: ' Then press ENTER or / for the decimal answer. Example 3: Find the value of the trigonometric ratio, sin ʹ (to 3 decimal places). On a Calculator page, complete the entry line as: sin (148 34') Then press ENTER. Chapter 9 Page 4 of 26

5 Adding and subtracting angles Angles can be added and subtracted using basic arithmetic. Like most other units of measurements, these operations can only be performed if the angles are measured in the same unit. To solve these problems without a calculator, separate the angles into degrees and minutes portions, perform each operation separately, and then combine the portions at the end. Example 4: (a) Add 46 o 37 and 65 o 49 (b) Subtract 16 o 55 from 40 o 20 Using a CAS calculator to add and subtract angles On a Calculator page, complete the entry lines as: 40 20' 16 55' 281 DMS 12 Press ENTER after each entry. Chapter 9 Page 5 of 26

6 Some geometry (angle) laws Chapter 9 Page 6 of 26

7 Example 5: Find the value of the pronumeral, f, the angle a beach umbrella makes on a level beach. Example 6: Find the value of the pronumerals A and C in the directions shown at right. Chapter 9 Page 7 of 26

8 9B Angles of Elevation and Depression Angle of Elevation The angle of elevation is the angle above the horizon or horizontal line. Angle of Depression The angle of depression is the angle below the horizon or horizontal line. Angles of elevation and depression are in a vertical plane. The angle of depression and angle of elevation shown below are alternate angles. Therefore Chapter 9 Page 8 of 26

9 Example 7: Find the angle of elevation (in degrees and minutes) of the tower measured from the road as given in the diagram. Example 8: Find the altitude of a plane (to the nearest metre) if the plane is sighted 4.5 km directly away from an observer who measures its angle of elevation as 26 23ʹ. Example 9: The angle of depression from the top of a 35-metre cliff to a house at the bottom is 23. How far from the base of the cliff is the house (to the nearest metre)? Chapter 9 Page 9 of 26

10 9C Bearings There are three main ways of specifying bearings or direction: 1. Standard compass bearings, for example, N, S, E, W, NE, SW etc 2. Other compass bearings, for example, N10 o W, S30 o E, N 45 o 37ʹ E etc 3. True bearings, for example, 100 o T, 297 o T, 045 o T, 056 o T etc Standard compass bearings There are 16 mains standard bearings as shown in the diagrams below. The N, S, E and W standard bearings are called cardinal points. Chapter 9 Page 10 of 26

11 Other compass bearings Often the direction required is not one of the 16 standard bearings. To specify other bearings the following approach is taken. 1. Start from north (N) or south (S). 2. Turn through the angle specified towards east (E) or west (W). Sometimes the direction may be specified unconventionally, for example, starting from east or west as given by the example W32 S. This bearing is equivalent to S58 W. True bearings True bearing is another method for specifying directions and is commonly used in navigation. To specify true bearings, first consider the following: 1. the angle is measured from north 2. the angle is measured in a clockwise direction to the bearing line 3. the angle of rotation may take any value from 0 to the symbol T is used to indicate it is a true bearing, for example, 125 T, 270 T 5. for bearings less than 100 T, use three digits with the first digit being a zero to indicate it is a bearing, for example, 045 T, 078 T. Chapter 9 Page 11 of 26

12 Example 10: specify the direction in the figure at right as: (a) a standard compass bearing (b) a compass bearing (c) a true bearing. Example 11: Draw a suitable diagram to represent the following directions. (a) S17 E (b) 252 T Chapter 9 Page 12 of 26

13 Example 12: Convert: (a) the true bearing, 137 T, to a compass bearing (b) the compass bearing, N25 W, to a true bearing. Example 13: Use your protractor to find the bearing of points A and B from location P. State the directions as true bearings and as compass bearings. Chapter 9 Page 13 of 26

14 9D Navigation and Specification of Locations When asked to solve problems, carefully draw a sketch of the situation. These sketches need to be converted to triangles with angles and lengths of sides included. We can then use Pythagoras theorem, trigonometric ratios, areas of triangles, similarity and sine or cosine rules. Hints 1. Carefully follow given instructions. 2. Always draw the compass rose at the starting point of the direction requested. 3. Key words are from and to. For example: The bearing from A to B (see diagram below left) is very different from the bearing from B to A (see diagram below right). 4. When you are asked to determine the direction to return directly back to an initial starting point, it is a 180 rotation or difference. For example, to return directly back after heading north, we need to change the direction to head south. Other examples are: Returning directly back after heading 135 T New bearing = = 315 T Returning directly back after heading 290 T New bearing = = 110 T Chapter 9 Page 14 of 26

15 Example 14: A ship leaves port, heading N30 E for 6 kilometres as shown. (a) How far north or south is the ship from its starting point (to 1 decimal place)? (b) How far east or west is the ship from its starting point (to 1 decimal place)? Chapter 9 Page 15 of 26

16 Example 15: A triangular paddock has two complete fences. From location D, one fence line is on a bearing of N23 W for 400 metres. The other fence line is S55 W for 700 metres. Find the length of fencing (to the nearest metre) required to complete the enclosure of the triangular paddock. Chapter 9 Page 16 of 26

17 Navigation Always sketch the journey and follow the steps below when answering navigational questions. 1. Draw a dot for your starting point and label it (A, B, home etc). 2. Draw a compass rose showing N, S, E, W on the dot and measure your bearing from NORTH. 3. Draw a line from your starting point to correspond with the bearing and distance. 4. At the end of the line put another dot and label it. 5. Draw a compass rose showing N, S, E, W on the second dot and measure your bearing from NORTH. 6. Draw a line from your second dot to correspond with the bearing and distance. Repeat steps 4-6 for further travel Key words to remember are from and to. A bearing from A to B A bearing to A from B Chapter 9 Page 17 of 26

18 Example 16: Soldiers on a reconnaissance set off on a return journey from their base camp. The journey consists of three legs. The first leg is on a bearing of 150 T for 3 km; the second is on a bearing of 220 T for 5km. Find the direction and distance of the third leg by which the group returns to its base camp. Chapter 9 Page 18 of 26

19 9E Triangulation - cosine and sine rules Triangulation should be used when: (a) the distance between two locations is given and (b) the direction from each of these two locations to the third inaccessible location is known. Example 17: How far (to 1 decimal place) is the fire from Tower A? Example 18: Two fire-spotting towers are 7 kilometres apart on an east west line. From Tower A a fire is seen on a bearing of 310 T. From Tower B the same fire is spotted on a bearing of N20 E. Which tower is closest to the fire and how far is that tower from the fire? Chapter 9 Page 19 of 26

20 Example 19: From the diagram at right find: (a) the length of CD (b) the bearing from C to D. Chapter 9 Page 20 of 26

21 9F Triangulation - Similarity Another method of solving triangulation problems is by using similar triangles. With these problems the scale factor is used to determine unknown lengths. 1. We need two corresponding lengths to establish the scale factor between the two similar triangles. A second accessible side will be used to scale up or down to the corresponding inaccessible side. 2. For similar triangles use the following rules as proof: (a) AAA all corresponding angles are the same (b) SSS all corresponding sides are in the same ratio (c) SAS two corresponding sides are in the same ratio with the same included angle. Example 20: Find the unknown length, x, from the pylon to the edge of the lake. Chapter 9 Page 21 of 26

22 9G Contour Maps A hill or raised region above sea level can be illustrated as a contour map Such maps are used: 1. by cross-country hikers in the sport of orienteering 2. by civil engineers planning new developments such as road constructions 3. as tourist information guides for local walks. These contour or topographic maps are a twodimensional overhead view of a region. Contour lines, along with a map scale, provide information about the region in a concise manner. Contour lines and intervals A contour line is defined as a line that joins places that are at the same height above sea level or a reference point. The distance between contour lines indicates the steepness of the slope. Chapter 9 Page 22 of 26

23 The closer the lines are the steeper the slope The further apart the contour lines are the slope the flatter the slope Intervals are indicated on the line to show the relative difference in height or altitude. Intervals change in regular multiples of 10 s or 100 s of metres Map scales Map scales are given as a ratio, for example 1: or a linear scale (below). The map ratio of a map is the ratio of a length on the map to the length it represents on the ground. For example: if two towns are 6km apart and on the map the distance is 6cm then the ratio is 6cm : 6 km 6cm : cm 1 : Therefore the map ratio is 1 : The scale is 1cm = 1km Chapter 9 Page 23 of 26

24 Average slope The average slope of land between two points A and B is given by: Average slope = gradient rise = run = tanθ A contour map can be used to draw a profile of the terrain. A profile is a side view of the land surface between two points, as shown above. Example 21: For the contour map, give an appropriate profile along the cross-sectional line AB. Chapter 9 Page 24 of 26

25 Example 22: From the given profile, construct an appropriate contour map. Chapter 9 Page 25 of 26

26 Example 23: For the contour map given, calculate: (a) the direct, straight-line distance between locations A and B (to the nearest metre) (b) the average slope of the land and the angle of elevation (in degrees and minutes) from the lower point to the upper one. Chapter 9 Page 26 of 26