Year 12 Further Mathematics UNIT 4 MODULE 2: Geometry and Trigonometry CHAPTER 8 - 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. Trigonometry, including: The solution of right-angled triangles using trigonometric ratios The solution of triangles using the sine and cosine rules Evaluation of areas of non-right-angled triangles using the formulas A = ½ absin(c) and A = s(s a)(s b)(s c). Question to complete 8A 3, 4, 5, 6, 7, 8, 9 8B 1(a, c, e), 3, 5, 6, 7, 8, 9, 10 8C 1, 2, 3, 4, 5, 8, 9, 10, 11 8D 1, 2, 3, 4, 5, 6, 7, 10, 11 8E 1, 2, 3, 4, 5, 8, 10, 12, 13, 15, 17 8F 1, 2, 3, 4, 5 8G 1, 2, 3, 4, 6, 8, 10, 11, 13, 14, 16, 8H 1, 2, 3, 4, 5, 6, 7 8I 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 Chapter 8 - Page 1 of 22
Table of Contents CHAPTER 8 - TRIGONOMETRY... 1 8A Pythagoras theorem... 3 Using a CAS calculator... 3 Using the CAS calculator... 4 8B Pythagorean triads... 5 How to generate a Pythagorean triad;... 5 8C Three- dimensional Pythagoras theorem... 6 Steps to solve 3- dimentional Pythagoras theorem... 6 Using a CAS calculator... 7 8D Trigonometric ratios... 8 How to label a right angled triangle... 8 Sine ratio (SOH)... 8 Cosine ratio (CAH)... 8 Tangent ratio (TOA)... 8 8E Introduction - Sine and Cosine rules... 11 Sine and cosine rules are designed to solve problems for non- right- angled triangles.... 11 How to label a triangle for the Sine and Cosine Rules... 11 The sine rule... 11 Using CAS calculator... 12 Using CAS calculator... 12 8F Ambiguous case of the sine rule.... 14 8G The cosine rule... 16 An unknown length when you have the lengths of two sides and the angle in between... 16 An unknown angle when you have the lengths of all three sides.... 17 8H Special Triangles... 19 Equilateral triangle... 19 Right- angled isosceles triangle... 19 8I Area of Triangles... 21 Method 1 - Area triangle = ½ Base Height... 21 Method 2 When you have two sides and the angle in between,... 21 Method 3 When you have all three sides we would use:... 21 Chapter 8 - Page 2 of 22
8A Pythagoras theorem Pythagoras theorem is used: (a) Only on right-angled triangles (b) To find an unknown length or distance, given the other two lengths. When using Pythagoras theorem: (a) Draw an appropriate diagram or sketch (b) Ensure the hypotenuse side, c, is opposite the right angle (90 ) (c) c 2 = a 2 + b 2 or shorter sides. c + 2 2 = a b, where c is the longest side or hypotenuse and a and b are the two To find one of the shorter sides (for example, side a), the formula transposes to: a 2 = c 2 b 2 and so a = c 2 b 2 Example 1: Find the length of the unknown side (to 1 decimal place) in the right-angled triangle shown. Using a CAS calculator On a Calculator page, press: MENU b 3: Algebra 3 1: Solve 1 Complete the entry line as: Solve (c 2 = a 2 + b 2, c) a = 4 and b = 7 Then press ENTER. For an approximate answer, press Ctrl / ENTER. (Note: Only the positive answer applies since c is a side length.) Or use nsolve, b36 Chapter 8 - Page 3 of 22
Example 2: Find the maximum horizontal distance (to the nearest metre) a ship could drift from its original anchored point, if the anchor line is 250 metres long and it is 24 metres to the bottom of the sea from the end of the anchor line on top of the ship s deck. It is important to sketch the diagram for the problem. Using the CAS calculator On a Calculator page, press: MENU b 3: Algebra 3 5: Solve 1 Complete the entry line as: Solve (c 2 = a 2 + b 2,a) c = 250 and b = 24 Then press Ctrl / ENTER. Only the positive solution applies. Chapter 8 - Page 4 of 22
8B Pythagorean triads A Pythagorean triad is a set of 3 numbers that satisfies Pythagoras theorem. (and hence, it is a right angled triangle) An example is the set of numbers 3, 4 and 5 c 2 = a 2 + b 2 5 2 = 3 2 + 4 2 25 = 9 + 16 Another example is a multiple of 2 of the above set: 6, 8 and 10 Others are 5, 12, 13 and 0.5, 1.2, 1.3 Example 3: Determine whether the following set of numbers 4, 6, 7 is a Pythagorean triad. How to generate a Pythagorean triad; 1. square an odd number (5 2 = ) 2. find 2 consecutive numbers that add up to the squared value ( + = 25) 3. the triad is the odd number you started with together with the 2 consecutive numbers (,, ) Try it with 7, 7 2 = 49, 24 + 25 =49, Test it does 7 2 + 24 2 = 25 2 Try it wth 9, Example 4: A triangle has sides of length 8 cm, 15 cm and 17 cm. Is the triangle right-angled? If so, where is the right angle? Chapter 8 - Page 5 of 22
8C Three- dimensional Pythagoras theorem Steps to solve 3- dimentional Pythagoras theorem To solve problems involving 3-dimentional Pythagoras theorem follow these steps: 1. Draw and label an appropriate diagram. 2. Identify the right-angled triangles that can be used to find unknown value(s). 3. To avoid rounding-errors use the surd form (eg 37 ) instead of 6.23 ). If the result is needed for another calculation. Example 5: To the nearest centimetre, what is the longest possible thin rod that could fit in the boot of a car? The boot can be modelled as a simple rectangular prism with the dimensions of 1.5 metres wide, 1 metre deep and 0.5 metres high. Chapter 8 - Page 6 of 22
Example 6: To find the height of a 100m square-based pyramid, with a slant height of 200m as shown, calculate the: (a) Length of AC (in surd form). (b) Length of AO (in surd form). (c) Height of the pyramid VO (to the nearest metre). Using a CAS calculator On a Calculator page, press: MENU b 3: Algebra 3 5: Solve 1 Complete the entry line as: solve(b 2 = a 2 + c 2, b) a = 100 and c = 100 Then press ENTER. AO is Half of b = Then complete the entry line as: solve(a 2 = v 2 + o 2, a) o = 200 and v = 50 2. Then press Ctrl / ENTER to get the decimal answer, or use nsolve. Note: Pressing ENTER will produce an approximate answer. Only the positive solution applies. Chapter 8 - Page 7 of 22
8D Trigonometric ratios Trigonometric ratios are used in right-angled triangles to find; 1. The length of one side, given an angle and length of another side 2. An angle, given the length of 2 sides. How to label a right angled triangle For the trigonometric ratios the following labelling convention should be applied: 1. The hypotenuse is opposite the right angle (90 ). 2. The opposite side is directly opposite the given angle, θ. 3. The adjacent side is next to the given angle, θ. Sine ratio (SOH) Cosine ratio (CAH) Tangent ratio (TOA) Using CAS calculator On a calculator page press the trig µkey, and select the function Chapter 8 - Page 8 of 22
Example 7: Find the length (to 1 decimal place) of the line AB. On a calculator page MENU b 3: Algebra 3 5: Solve 1 Complete the entry line as: nsolve sin(50 ) = o 15, o Hint: press the trig µkey, and select sin Hint: press ¹or º to get o (degrees) Example 8: Find the length of the guy wire (to the nearest centimetre) supporting a flagpole, if the angle of the guy wire to the ground is 70 o and it is 2 metres from the base of the flagpole. Chapter 8 - Page 9 of 22
Example 9: Find the length of the shadow (to 1 decimal place) cast by a 3-metre pole when the angle of the sun to the horizontal is 70 o. Example 10: Find the smallest angle (to the nearest degree) in a 3, 4, 5 Pythagorean triangle. Chapter 8 - Page 10 of 22
8E Introduction - Sine and Cosine rules Sine and cosine rules are designed to solve problems for non- right- angled triangles. How to label a triangle for the Sine and Cosine Rules For the sine and cosine rules the following labelling convention should be used. Angle A is opposite side a (at vertex A) Angle B is opposite side b (at vertex B) Angle C is opposite side c (at vertex C) The sine rule The sine rule is used to find unknown lengths and angles of non-right-angled triangles if you are given 1. Two angles and one side 2. An angle and its opposite length and one other side The sine rule states that for a triangle ABC (shown below) a sin A = b sin B = c sin C Use for working out the unknown length sin A a sin B = b = sin C Use for working out the unknown angle c Chapter 8 - Page 11 of 22
Example 11: Find the unknown length x cm in the triangle below. Using CAS calculator MENU b 3: Algebra 3 5: Numerical Solve 6 Complete the entry line as:! nsolve =!, b!"#!"#!"#!" Then press ENTER. Example 12: Find the unknown length, x cm (to 2 decimal places). Using CAS calculator MENU b 3: Algebra 3 5: Numerical Solve 6 Complete the entry line as:! nsolve =!, c!"#!!!!"#!" Then press ENTER. Chapter 8 - Page 12 of 22
Example 13: For a triangle PQR, find the unknown angle, P (to the nearest degree). When calculating the size of an angle on a CAS calculator, it is a good idea to instruct the calculator to calculate all angles between 0 and 180. To do this on a Calculator page, MENU b 3: Algebra 3 5: nsolve 6 Complete the entry line as: Solve! =!!"#!!"#!", p 0 p 180 Then press ENTER. Example 14: A compass used for drawing circles has two equal legs joined at the top. The legs are 8 centimetres long. If it is opened to an included angle of 36 degrees between the two legs, find the radius of the circle that would be drawn (to 1 decimal place). On a Calculator page, MENU b 3: Algebra 3 5: nsolve 6 Complete the entry line as:! nsolve =!, b!"#!"!"#!" Then press ENTER. Chapter 8 - Page 13 of 22
8F Ambiguous case of the sine rule. On your calculator, investigate the values for each of these pairs of sine ratios: sin 30 = and sin 150 = sin 110 = and sin 70 =. The calculator will give only the acute angle not the obtuse angle The situation is illustrated practically in the diagram below where the sine of the acute angle equals the sine of the obtuse angle. Therefore always check your diagram to see if the unknown angle is the acute or obtuse angle or perhaps either. Another situation is illustrated in the two diagrams below. The triangles have two corresponding sides equal, a and b, as well as angle B. The sine of 110 = sine of 70 but side c is very different. This ambiguity occurs when the smaller known side is opposite the known angle. B B c = 9cm a = 6cm c = 9 cm a = 6 cm A 34 o b C A 34 o b C obtuse angle = 180 acute angle Example 15: To the nearest degree, find the angle, U, in a triangle, given t = 7, u = 12 and angle T is 25 o. Chapter 8 - Page 14 of 22
Using the CAS calculator When calculating the size of an angle on a CAS calculator, it is a good idea to instruct the calculator to calculate all angles between 0 and 180. To do this on a Calculator page, complete the entry line as: solve!" =!!"#(!)!"#!", u 0 u 180 Then press Ctrl / ENTER. Example 16: In the obtuse-angled triangle PQR, find the unknown angle (to the nearest degree), P. Chapter 8 - Page 15 of 22
8G The cosine rule Cosine rule is used to find: An unknown length when you have the lengths of two sides and the angle in between Chapter 8 - Page 16 of 22
An unknown angle when you have the lengths of all three sides. Chapter 8 - Page 17 of 22
Example 17: Find the unknown length (to 2 decimal places), x, in the triangle below. Example 18: Find the size of angle x in the triangle below, to the nearest degree. Chapter 8 - Page 18 of 22
8H Special Triangles Equilateral triangle 60 o Equilateral triangles have three equal sides and three equal angles. The three angles are always 60. Right- angled isosceles triangle 45 o a c = 2 a 45 o a Right-angled isosceles triangles have one right angle (90 ) opposite the longest side (hypotenuse) and two equal sides and angles. The two other angles are always 45. The hypotenuse is always 2 times the length of the smaller sides. Chapter 8 - Page 19 of 22
Example 19: Find the values of r and angle θ in the hexagon. Example 20: Find the value of the pronumeral (to 1 decimal place) in the figure. Chapter 8 - Page 20 of 22
8I Area of Triangles Three possible methods can be used to find the area of a triangle: Method 1 - Areatriangle = ½ Base Height A = 2 1 bh Method 2 When you have two sides and the angle in between, A = 2 1 ab sin C Method 3 When you have all three sides we would use: A = s( s a)( s b)( s c) where a + b + c s = (s = semi-perimeter) 2 Example 21: Find the area of the triangle Method 1: Chapter 8 - Page 21 of 22
Example 22: Find the area of the triangle (to 2 decimal places). Example 23: Find the area of a triangle PQR (to 1 decimal place), given p = 6, q = 9 and r = 4, with measurements in centimetres. Open a Calculator page. An alternative variable name for Area is needed as A cannot be used if lowercase a is used for length. Complete the entry line as: Solve ( m s ( s a) ( s b) ( s c) ) = Then press Ctrl / ENTER. s = a + b + c 2 Chapter 8 - Page 22 of 22