Trigonometry I PDF - Jacaranda Maths Quest 10

9 Trigonometry I LESSON SEQUENCE 9.1 Overview...............................................................................................................................................................440 9.2 Pythagoras’ theorem..................................................................

9 Trigonometry I LESSON SEQUENCE 9.1 Overview...............................................................................................................................................................440 9.2 Pythagoras’ theorem....................................................................................................................................... 443 9.3 Pythagoras’ theorem in three dimensions (Path).................................................................................. 451 9.4 Trigonometric ratios........................................................................................................................................ 457 9.5 Using trigonometry to calculate side lengths.........................................................................................463 9.6 Using trigonometry to calculate angle size............................................................................................. 468 9.7 Angles of elevation and depression........................................................................................................... 473 9.8 Bearings................................................................................................................................................................478 9.9 Applications of trigonometry........................................................................................................................ 485 9.10 Review................................................................................................................................................................... 490 LESSON 9.1 Overview Why learn this? Nearly 2000 years ago Ptolemy of Alexandria published the first book of trigonometric tables, which he used to chart the heavens and plot the courses of the Moon, stars and planets. He also created geographical charts and provided instructions on how to create maps. The word trigonometry is derived from Greek words ‘trigonon’ and ‘metron’ meaning triangles and measure respectively, and reportedly has been studied since the third century BCE. This field of mathematics was studied across the world, with major discoveries made in India, China, Greece and Persia, to name a few. The works ranged from developing relationships, axioms and proofs to its application to everyday use and life. Trigonometry is the branch of mathematics that makes the whole universe more easily understood. The role and use of trigonometry in navigation in the early years were crucial and its application and study grew from there. Today, it is used in architecture, surveying, astronomy and, as previously mentioned, navigation. It also provides the foundation of the study of sound and light waves, resulting in its application in the areas of music manufacturing and composition, study of tides, radiology and many other fields. Hey students! Bring these pages to life online Watch Engage with Answer questions videos interactivities and check solutions Find all this and MORE in jacPLUS Reading content Extra learning and rich media, resources including interactivities and videos for Differentiated every concept question sets Questions with immediate feedback, and fully worked solutions to help students get unstuck 440 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition Exercise 9.1 Pre-test 1. Determine the value of the pronumeral w, correct to 2 decimal places. 7.3 cm w cm 2.41 cm 2. Determine the value of x, correct to 2 decimal places. 5 cm x cm 2x cm 3. PATH A square-based pyramid is 16 cm high. Each sloping edge is 20 cm long. Calculate the length of the sides of the base, in cm correct to 2 decimal places. 4. PATH A cork is in the shape of a truncated cone; both the top and the base of the cork are circular. 20 cm 32 cm 30 cm Calculate the sum of diameters of the top and the base. Give your answer in cm to 2 decimal places. 5. Evaluate sin(20°37′) correct to 4 decimal places. 6. Calculate the size of the angle 𝜃, correct to the nearest minute, given that cos(𝜃) = 0.5712. Give your answer in degrees and minutes. 7. Determine the size of the angle 𝜃, correct to the nearest minute. 2.4 θ 3.2 TOPIC 9 Trigonometry I 441  8. Calculate y, correct to 1 decimal place. ym 28° 42' 11.8 m 9. MC Tyler is standing 12 m from a flagpole and measures the angle of elevation from his eye level to the top of the pole as 62°. The distance from Tyler’s eyes to the ground is 185 cm. The height of the flagpole correct to 2 decimal places is: A. 20.71 m B. 22.56 m C. 24.42 m D. 207.56 m 10. Change each of the following compass bearings to true bearings. a. N20°E b. S47°W c. N33°W d. S17°E 11. MC A boat travels 15 km from A to B on a bearing of 032°T. The bearing from B to A is: A. 032°T B. 058°T C. 122°T D. 212°T 12. A bushwalker travels N50°W for 300 m and then changes direction 220°T for 0.5 km. Determine how many metres west the bushwalker is from his starting point. Give your answer in km correct to 1 decimal place. 13. MC P and Q are two points on a horizontal line that are 120 metres apart. If the angles of elevation from P and Q to the top of the mountain are 34°5′ and 41°16′ respectively, the height of the mountain correct to 1 decimal place is: A. 81.2 metres B. 105.3 metres C. 120.5 metres D. 354.7 metres 14. Determine the value of x in the following figure, correct to 1 decimal place. 45° 25° 13' 8 x 15. PATH MC In a right square-based pyramid, the square base has a length of 7.2 cm. If the angle between the triangular face and the base is 55°, the angle the sloping edge makes with the base is: 55° 7.2 cm A. 45.3° B. 55° C. 63.4° D. 47.7° 442 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition  LESSON 9.2 Pythagoras’ theorem LEARNING INTENTION At the end of this lesson you should be able to: identify similar right-angled triangles when corresponding sides are in the same ratio and corresponding angles are congruent apply Pythagoras’ theorem to calculate the third side of a right-angled triangle when two other sides are known. 9.2.1 Similar right-angled triangles eles-4799 D Two similar right-angled triangles have the same angles when the corresponding sides are in the same ratio. The hypotenuse is the longest side of a right-angled triangle and is always the side that is opposite the right angle. A 6 cm 10 cm The corresponding sides are in the same ratio. = = 3 cm 5 cm AB AC BC DE DF EF B 4 cm C E 8 cm F To write this using the side lengths of the triangles = = gives: AB 3 1 DE 6 2 = = AC 5 1 DF 10 2 = = BC 4 1 EF 8 2 This means that for right-angled triangles, when the angles are fixed, the ratios of the sides in the triangle are constant. We can examine this idea further by completing the following activity. Using a protractor and ruler, draw an angle of 70° measuring horizontal distances of 3 cm, 7 cm and 10 cm as demonstrated c in the diagram below. b Note: Diagram not drawn to scale. a a ≈ 8.24 cm, b ≈ 19.23 cm, c ≈ 27.47 cm Measure the perpendicular heights a, b and c. 70° 3 cm 7 cm 10 cm TOPIC 9 Trigonometry I 443  To test if the theory for right-angled triangles, that when the angles are fixed the ratios of the sides in the triangle are constant, is correct, calculate the ratios of the side lengths. ≈ ≈ 2.75, ≈ ≈ 2.75, ≈ ≈ 2.75 a 8.24 b 19.23 c 27.47 3 3 7 7 10 10 The ratios are the same because the triangles are similar. This important concept forms the basis of trigonometry. 9.2.2 Review of Pythagoras’ theorem eles-4800 Pythagoras’ theorem gives us a way of finding the length of the third side in a right angle triangle, if we know the lengths of the two other sides. Pythagoras’ theorem Pythagoras’ theorem: c2 = a2 + b2 Determining the hypotenuse: c = a2 + b2 c √ a Determining one of the two shorter sides: a = c2 − b2 or b = c2 − a2 √ √ b WORKED EXAMPLE 1 Calculating the hypotenuse For the following triangle, calculate the length of the hypotenuse x, correct to 1 decimal place. x 5 8 THINK WRITE/DRAW 1. Copy the diagram and label the sides a, b and c. Remember to label the hypotenuse as c. a=5 c=x b=8 2. Write Pythagoras’ theorem. c2 = a2 + b2 x2 = 52 + 82 = 25 + 64 3. Substitute the values of a, b and c into this rule and simplify. = 89 x = ± 89 correct to 1 decimal place, since x > 0. x ≈ 9.4 √ 4. Take the square root of both sides. Round the positive answer 444 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition WORKED EXAMPLE 2 Calculating the length of the shorter side Calculate the length, correct to 1 decimal place, of the unmarked side of the following triangle. 14 cm 8 cm THINK WRITE/DRAW 1. Copy the diagram and label the sides a, b and c. Remember to label the hypotenuse as c; it does not matter which side is a a and which side is b. c = 14 b=8 2. Write Pythagoras’ theorem. c2 = a2 + b2 142 = a2 + 82 196 = a2 + 64 3. Substitute the values of a, b and c into this rule and solve for a. a2 = 196 − 64 = 132 a = ± 132 round to 1 decimal place (a > 0). ≈ 11.5 cm √ 4. Evaluate a by taking the square root of both sides and WORKED EXAMPLE 3 Solving a practical problem using Pythagoras’ theorem A ladder that is 5.5 m long leans up against a vertical wall. The foot of the ladder is 1.5 m from the wall. Determine how far up the wall the ladder reaches. Give your answer in metres correct to 1 decimal place. THINK WRITE/DRAW 1. Draw a diagram and label the sides a, b and c. Remember to label the hypotenuse as c. c = 5.5 m a b = 1.5 m 2. Write Pythagoras’ theorem. c2 = a2 + b2 5.52 = a2 + 1.52 30.25 = a2 + 2.25 3. Substitute the values of a, b and c into this rule and simplify. a2 = 30.25 − 2.25 = 28 a = ± 28 1 decimal place, a > 0. ≈ 5.3 √ 4. Evaluate a by taking the square root of 28. Round to 5. Write the answer in a sentence. The ladder reaches 5.3 m up the wall. TOPIC 9 Trigonometry I 445  WORKED EXAMPLE 4 Determining the unknown sides Determine the unknown side lengths of the triangle, correct to 2 decimal places. 3x 78 m 2x THINK WRITE/DRAW 1. Copy the diagram and label the sides a, b and c. b = 3x c = 78 m a = 2x 2. Write Pythagoras’ theorem. c2 = a2 + b2 782 = (3x)2 + (2x)2 6084 = 9x2 + 4x2 3. Substitute the values of a, b and c into this rule and simplify. 6084 = 13x2 4. Rearrange the equation so that the pronumeral is on the left-hand side 13x2 = 6084 of the equation. = 13x2 6084 5. Divide both sides of the equation by 13. x2 = 468 13 13 x = ± 468 ≈ 21.6333 √ 6. Evaluate x by taking the square root of both sides. Round the answer correct to 2 decimal places. 2x ≈ 43.27m 3x ≈ 64.90 m 7. Substitute the value of x into 2x and 3x to determine the lengths of the unknown sides. COMMUNICATING — COLLABORATIVE TASK: History of Pythagoras’ theorem Pythagoras’ theorem was known about before the age of Pythagoras. In small groups, research which other civilisations knew about the theory and construct a timeline as a class for its history. Resources Resourceseses Interactivities Finding a shorter side (int-3845) Finding the hypotenuse (int-3844) 446 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition Exercise 9.2 Pythagoras’ theorem 9.2 Quick quiz 9.2 Exercise Individual pathways PRACTISE CONSOLIDATE MASTER 1, 5, 7, 10, 11, 14, 17, 21, 22, 25 2, 3, 8, 12, 15, 19, 20, 23, 26 4, 6, 9, 13, 16, 18, 24, 27 Fluency 1. WE1 For each of the following triangles, calculate the length of the hypotenuse, giving answers correct to 2 decimal places. a. 4.7 b. 19.3 c. 804 6.3 27.1 562 2. For each of the following triangles, calculate the length of the hypotenuse, giving answers correct to 2 decimal places. a. b. 0.9 c. 152 7.4 87 10.3 2.7 3. WE2 Determine the value of the pronumeral, correct to 2 decimal places. a. s b. c. u 1.98 8.4 30.1 47.2 2.56 17.52 t 4. Determine the value of the pronumeral, correct to 2 decimal places. a. 0.28 b. 2870 c. v 468 1920 x w 0.67 114 TOPIC 9 Trigonometry I 447  5. WE3 The diagonal of a rectangular NO SMOKING sign is 34 cm. If the height of this sign is 25 cm, calculate the width of the sign, in cm correct to 2 decimal places. 6. A right-angled triangle has a base of 4 cm and a height of 12 cm. Calculate the length of the hypotenuse in cm correct to 2 decimal places. 7. Calculate the lengths of the diagonals (in cm to 2 decimal places) of squares that have side lengths of: a. 10 cm b. 17 cm c. 3.2 cm. 8. The diagonal of a rectangle is 90 cm. One side has a length of 50 cm. Determine, correct to 2 decimal places: a. the length of the other side b. the perimeter of the rectangle c. the area of the rectangle. 9. WE4 Determine the value of the pronumeral, correct to 2 decimal places for each of the following. a. b. c. 2x 3x 3x 25 4x 18 6x 30 x Understanding 10. An isosceles triangle has a base of 25 cm and a height of 8 cm. Calculate the length of the two equal sides, in cm correct to 2 decimal places. 11. An equilateral triangle has sides of length 18 cm. Determine the height of the triangle, in cm correct to 2 decimal places. 12. A right-angled triangle has a height of 17.2 cm, and a base that is half the height. Calculate the length of the hypotenuse, in cm correct to 2 decimal places. 13. The road sign shown is based on an equilateral triangle. Determine the height of the sign and, hence, calculate its area. Round your answers to 2 decimal places. 14. A flagpole, 12 m high, is supported by three wires, attached from the top of the pole to the ground. Each wire is pegged into the ground 5 m from 84 cm the pole. Determine how much wire is needed to support the pole, correct to the nearest metre. 15. Sarah goes canoeing in a large lake. She paddles 2.1 km to the north, then 3.8 km 3.8 km to the west. Use the triangle shown to determine how far she must then paddle to get back to her starting point in the shortest possible way, in km correct to 2 decimal places. 2.1 km Starting point 448 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition 16. A baseball diamond is a square of side length 27 m. When a runner on first base tries to steal second base, the catcher has to throw the ball from home base to second base. Calculate the distance of the throw, in metres correct to 1 decimal place. Second base 27 m First base Home base Catcher 17. A rectangle measures 56 mm by 2.9 cm. Calculate the length of its diagonal in millimetres correct to 2 decimal places. 18. A rectangular envelope has a length of 24 cm and a diagonal measuring 40 cm. Calculate: a. the width of the envelope, correct to the nearest cm b. the area of the envelope, correct to the nearest cm2. 19. A swimming pool is 50 m by 25 m. Peter is bored by his usual training routine, and decides to swim the diagonal of the pool. Determine how many diagonals he must swim to complete his normal distance of 1500 m. 20. A hiker walks 2.9 km north, then 3.7 km east. Determine how far in metres she is from her starting point. Give your answer in metres to 2 decimal places. 21. A square has a diagonal of 14 cm. Calculate the length of each side, in cm correct to 2 decimal places. Communicating, reasoning and problem solving 22. The triangles below are right-angled triangles. Two possible measurements have been suggested for the hypotenuse in each case. For each triangle, complete calculations to determine which of the lengths is correct for the hypotenuse in each case. Show your working. a. b. c. 60 or 65 273 33 185 or 195 175 305 or 308 56 136 60 TOPIC 9 Trigonometry I 449 23. The square root of a number usually gives us both a positive and negative answer. Explain why we take only the positive answer when using Pythagoras’ theorem. 24. Four possible side length measurements are 105, 208, 230 and 233. Three of them together produce a right-angled triangle. a. Explain which of the measurements could not be the hypotenuse of the triangle. b. Complete as few calculations as possible to calculate which combination of side lengths will produce a right-angled triangle. 25. The area of the rectangle MNPQ is 588 cm2. Angles MRQ M 28 cm N y cm and NSP are right angles. R a. Determine the integer value of x. b. Determine the length of MP. x cm c. Calculate the value of y and hence determine the length of RS, in cm correct to 1 decimal place. S y cm Q P 26. Triangle ABC is an equilateral triangle of side length x cm. Angles ADB and DBE are right angles. Determine the value of x in cm, correct to 2 decimal places. B 16 cm E x cm 20 cm A D C 27. The distance from Earth to the Moon is approximately 385 000 km and the distance from Earth to the Sun is approximately 147 million kilometres. In a total eclipse of the Sun, the moon moves between the Sun and Earth, thus blocking the light of the Sun from reaching Earth and causing a total eclipse of the Sun. If the diameter of the Moon is approximately 3474 km, evaluate the diameter of the Sun. Express your answer to the nearest 10 000 km. 14 70 00 00 0k m 385 000 km 450 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition  LESSON 9.3 Pythagoras’ theorem in three dimensions (Path) LEARNING INTENTION At the end of this lesson you should be able to apply Pythagoras’ theorem to: determine unknown lengths when a 3D diagram is given determine unknown lengths in situations by first drawing a diagram. 9.3.1 Applying Pythagoras’ theorem in three dimensions eles-4801 Many real-life situations involve 3-dimensional (3-D) objects: objects with length, width and height. Some common 3-D objects used in this section include cuboids, pyramids and right-angled wedges. Cuboid Pyramid Right-angled wedge In diagrams of 3-D objects, right angles may not look like right angles, so it is important to redraw sections of the diagram in two dimensions, where the right angles can be seen accurately. WORKED EXAMPLE 5 Applying Pythagoras’ theorem to 3D objects Determine the length AG in this rectangular prism (cuboid), in cm correct to 2 decimal places. A B 5 cm C D F E 9 cm H 10 cm G THINK WRITE/DRAW 1. Draw the diagram in three dimensions. A B ∠AEG is a right angle. Draw the lines AG and EG. 5 cm C D F E 9 cm H 10 cm G 2. Draw ΔAEG, showing the right angle. Only 1 side is known, so A EG must be found. 5 E G TOPIC 9 Trigonometry I 451  3. Draw EFGH in two dimensions and label the diagonal EG as x. E F x 9 9 H G 4. Use Pythagoras’ theorem to calculate x where x > 0 x2 = 92 + 102 10 = 81 + 100 = 181 since length is positive. x = 181 √ 5. Place this information on triangle AEG. Label the side AG as y. A y 5 E G √181 6. Use Pythagoras’ theorem to calculate y where y > 0 y2 = 52 + (√ )2 = 25 + 181 181 = 206 since length is positive. y = ± 206 ≈ 14.35 √ 7. Write the answer in a sentence. The length of AG is 14.35 cm. WORKED EXAMPLE 6 Drawing a diagram to solve problems A piece of cheese in the shape of a right-angled wedge sits on a table. It has a rectangular base measuring 14 cm by 8 cm, and is 4 cm high at the thickest point. An ant crawls diagonally across the sloping face. Determine how far, to the nearest millimetre, the ant walks. THINK WRITE/DRAW 1. Draw a diagram in three dimensions and label the B C dotted line. ∠BED is a right angle. vertices. Mark BD, the path taken by the ant, with a E x 4 cm F 8 cm A 14 cm D 2. Draw ΔBED, showing the right angle. Only one side is B known, so ED must be found. 4 D E 452 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition 3. Draw EFDA in two dimensions, and label the E F diagonal ED. Label the side ED as x. 8 x 8 A D c2 = a2 + b2 14 x > 0 since length is positive. 4. Use Pythagoras’ theorem to calculate x where x2 = 82 + 142 = 64 + 196 = 260 x = 260 √ 5. Place this information on triangle BED. Label B the side BD as y. y 4 D E √260 y2 = 42 + (√ )2 y > 0 since length is positive. 6. Use Pythagoras’ theorem to calculate y where 260 = 16 + 260 = 276 y = 276 √ ≈ 16.61 cm ≈ 166.1 mm 7. Write the answer in a sentence. The ant walks 166 mm, correct to the nearest millimetre. DISCUSSION Look around the room you are in. How many right angles can you spot in three-dimensional objects? Make a list of them and compare your list to that of another student. Resources Resourceseses Video eLesson Pythagoras’ theorem in three dimensions (eles-1913) Interactivity Right angles in 3-dimensional objects (int-6132) TOPIC 9 Trigonometry I 453 Exercise 9.3 Pythagoras’ theorem in three dimensions (Path) 9.3 Quick quiz 9.3 Exercise Individual pathways PRACTISE CONSOLIDATE MASTER 1, 4, 7, 11, 12, 15 2, 5, 8, 13, 16 3, 6, 9, 10, 14, 17 Where appropriate in this exercise, give answers correct to 2 decimal places. Fluency 1. WE5 Calculate the length of AG in each of the following figures. a. A B b. A B c. A B D C C 8 C D D 10.4 12 E E F F 8 E F 9.2 5 H 8 G H H 11.5 G 5 G 2. Consider the wedge shown. Calculate the length of CE in the wedge and, hence, obtain the length of AC. A B E 4 F D C 7 10 3. If DC = 3.2 m, AC = 5.8 m, and CF = 4.5 m in the figure, calculate the length of AD and BF. A B F D C 4. Consider the pyramid shown. Calculate the length of BD and, hence, the height of the pyramid. V 6 A B 6 D 6 C 5. The sloping side of a cone is 16 cm and the height is 12 cm. Determine the length of the radius of the base. 12 cm 16 cm r 454 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition  6. The pyramid ABCDE has a square base. The pyramid is 20 cm high. Each sloping edge measures 30 cm. Calculate the length of the sides of the base. E EM = 20 cm A B M D C Understanding 7. WE6 A piece of cheese in the shape of a right-angled wedge sits on B C a table. It has a base measuring 20 mm by 10 mm, and is 4 mm high 4 mm at the thickest point, as shown in the figure. A fly crawls diagonally E across the sloping face. Determine how far, to the nearest millimetre, F 10 mm the fly walks. A 20 mm D 8. A 7 m high flagpole is in the corner of a rectangular park that measures 200 m by 120 m. Give your answers to the following questions correct to 7m 2 decimal places. 200 m A a. Calculate: i. the length of the diagonal of the park 120 m ii. the distance from A to the top of the pole B iii. the distance from B to the top of the pole. b. A bird flies from the top of the pole to the centre of the park. Calculate how far it flies. 9. A candlestick is in the shape of two cones, joined at the vertices as shown. The smaller cone has a diameter and sloping side of 7 cm, and the larger one has a diameter and sloping side of 10 cm. Calculate the total height of the candlestick. 10. The total height of the shape below is 15 cm. Calculate the length of the sloping side of the pyramid. 15 cm 5 cm 11 cm 11 cm TOPIC 9 Trigonometry I 455 11. A sandcastle is in the shape of a truncated cone as shown. Calculate the length of 20 cm the diameter of the base. 30 cm 32 cm Communicating, reasoning and problem solving 12. Stephano is renovating his apartment, which he accesses through two corridors. The corridors of the apartment building are 2 m wide with 2-m-high ceilings, and the first corridor is at right angles to the second. Show that he can carry lengths of timber up to 6 m long to his apartment. 13. The Great Pyramid in Egypt is a square-based pyramid. The square base has a side length of 230.35 m and the perpendicular height is 146.71 m. Determine the slant height, s, of the great pyramid. Give s your answer correct to 1 decimal place. 146.71 m 230.35 m 14. A tent is in the shape of a triangular prism, with a height of 140 cm as 140 cm shown in the diagram. The width across the base of the door is 1 m and the tent is 2.5 m long. a. Calculate the length of each sloping side, in metres. b. Using your answer from part a calculate the area of fabric used in the 2.5 m construction of the sloping rectangles which form the sides. Show full 1m working out. 32 cm × 15 cm × 4 cm. Ignore the thickness of the steel rod. 15. Determine the exact length of the longest steel rod that can sit inside a cuboid with dimensions 16. Angles ABD, CBD and ABC are right angles. Determine the value of h, D correct to 3 decimal places. 35 h 37 B A 36 C 17. The roof of a squash centre is constructed to allow for maximum use of sunlight. Determine the value of h, giving your answer correct to 1 decimal place. 57.08 m y x h 35 m x 20 m 456 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition  LESSON 9.4 Trigonometric ratios LEARNING INTENTION At the end of this lesson you should be able to: calculate the values of sin(𝜃), cos(𝜃) and tan(𝜃) for different values of 𝜃 identify and define the trigonometric ratios for right-angled triangles and use trigonometric notation express angles in degrees, minutes and seconds. 9.4.1 Trigonometric ratios eles-6274 If one of the two acute angles is named (for example, 𝜃), then the other two sides can also be given names, In a right-angled triangle, the longest side is called the hypotenuse. as shown in the following diagram. Trigonometric ratios Opposite Hypotenuse 𝜃 Adjacent Using the diagram, the following three trigonometric ratios can be defined. The sine ratio: sine(𝜃) = H O length of Opposite θ length of Hypotenuse O The cosine ratio: H cosine(𝜃) = θ length of Adjacent θ A length of Hypotenuse SOH A The tangent ratio: tangent(𝜃) = length of Opposite TOA CAH length of Adjacent The names of the three ratios are usually shortened to sin(𝜃), cos(𝜃) and tan(𝜃). Sin(𝜃) = Opposite over Hypotenuse and so on. The three ratios are often remembered using the mnemonic SOHCAHTOA, where SOH means Calculating trigonometric values using a calculator Traditionally angles were measured in degrees, minutes and seconds, where 60 seconds = 1 minute and The sine, cosine and tangent of an angle have numerical values that can be found using a calculator. 60 minutes = 1 degree. This is known as a sexagesimal system as the division are based on 60. For example, 50°33′ 48′′ means 50 degrees, 33 minutes and 48 seconds. In this topic, you will be asked to give answers in degrees and minutes only. TOPIC 9 Trigonometry I 457  WORKED EXAMPLE 7 Calculating values (ratios) from angles Calculate the value of each of the following, correct to 4 decimal places, using a calculator. (Remember to first work to 5 decimal places before rounding.) ′ a. cos(65°57′ ) b. tan(56°45 ) a. cos(65°57′) ≈ 0.407 53 THINK WRITE ≈ 0.4075 a. Write your answer to the required number of decimal places. b. tan(56°45′) ≈ 1.525 253 5 ≈ 1.5253 b. Write your answer to the correct number of decimal places. WORKED EXAMPLE 8 Calculating angles from ratios Calculate the size of angle 𝜃, correct to the nearest degree, given sin(𝜃) = 0.7854. sin(𝜃) = 0.7854 THINK WRITE 1. Write the given equation. 𝜃 = sin−1 (0.7854) −1 ≈ 51.8° 2. To calculate the size of the angle, we need to undo sine with its inverse, sin. (Ensure your calculator is in degrees mode.) 3. Write your answer to the nearest degree. 𝜃 ≈ 52° WORKED EXAMPLE 9 Expressing angles in degrees and minutes Calculate the value of 𝜃: a. correct to the nearest minute, given that cos(𝜃) = 0.2547 b. correct to the nearest minute, given that tan(𝜃) = 2.364. a. cos(𝜃) = 0.2547 THINK WRITE a. 1. Write the equation. cos−1 (0.2547) ≈ 75°14′39′′ ≈ 75°15′ 2. Write your answer, including seconds. There are 60 seconds in 1 minute. Round (Remember 60′′ = 1′, so 39′′ is rounded up.) to the nearest minute. b. 1. Write the equation. b. tan(𝜃) = 2.364 tan−1 2.364 ≈ 67°4′15.8′′ ≈ 67°4′ 2. Write the answer, rounding to the nearest minute. 458 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition WORKED EXAMPLE 10 Expressing trigonometric ratios as equations Write the equation that relates the two marked sides and the marked angle. a. b. 22 12 8 x 40° b THINK WRITE/DRAW a. 1. Label the given sides of the triangle. a. 12 = H 8=O b sin(𝜃) = O 2. Write the ratio that contains O and H. O = 8, H = 12 H 3. Identify the values of the pronumerals. sin(b) = = 8 2 4. Substitute the values of the pronumerals into the ratio and simplify the fraction. (Since 12 3 replace 𝜃 with b.) the given angle is denoted with the letter b, b. 1. Label the given sides of the triangle. b. 22 = A x=O 40° tan(𝜃) = O 2. Write the ratio that contains O and A. O = x, A = 22, 𝜃 = 40° A 3. Identify the values of the pronumerals. tan(40°) = x 4. Substitute the values of the pronumerals into the ratio. 22 DISCUSSION Can you think of a real-world situation where you could use trigonometric ratios to solve a problem? Resources Resourceseses Interactivity Trigonometric ratios (int-2577) TOPIC 9 Trigonometry I 459 Exercise 9.4 Trigonometric ratios 9.4 Quick quiz 9.4 Exercise Individual pathways PRACTISE CONSOLIDATE MASTER 1, 5, 7, 9, 11, 17, 22 2, 6, 12, 15, 18, 21 3, 4, 8, 10, 13, 14, 16, 19, 20, 23 Fluency 1. Calculate each of the following, correct to 4 decimal places. a. sin(30°) b. cos(45°) c. tan(25°) d. sin(57°) e. tan(83°) f. cos(44°) WE7 For questions 2 to 4, calculate each of the following, correct to 4 decimal places. 2. a. sin(40°30′) b. cos(53°57′) c. tan(27°34′) d. tan(123°40′) e. sin(92°32′) f. sin(42°8′) 3. a. cos(35°42′) b. tan(27°42′) c. cos(143°25′) d. sin(23°58′) e. cos(8°54′) 4. a. sin(286°) b. tan(420°) c. cos(845°) d. sin(367°35′) Calculate the size of angle 𝜃, correct to the nearest degree, for each of the following. a. sin(𝜃) = 0.763 b. cos(𝜃) = 0.912 c. tan(𝜃) = 1.351 5. WE8 6. Calculate the size of angle 𝜃, correct to the nearest degree, for each of the following. a. cos(𝜃) = 0.321 b. tan(𝜃) = 12.86 c. cos(𝜃) = 0.756 Calculate the size of the angle 𝜃, correct to the nearest minute. a. sin(𝜃) = 0.814 b. sin(𝜃) = 0.110 c. tan(𝜃) = 0.015 7. WE9a 8. Calculate the size of the angle 𝜃, correct to the nearest minute. a. cos(𝜃) = 0.296 b. tan(𝜃) = 0.993 c. sin(𝜃) = 0.450 Calculate the size of the angle 𝜃, correct to the nearest minute. a. tan(𝜃) = 0.5 b. cos(𝜃) = 0.438 c. sin(𝜃) = 0.9047 9. WE9b 10. Calculate the size of the angle 𝜃, correct to the nearest minute. a. tan(𝜃) = 1.1141 b. cos(𝜃) = 0.8 c. tan(𝜃) = 43.76. For questions 11 to 13, calculate the value of each expression, correct to 3 decimal places. 2 11. a. 3.8 cos(42°) b. 118 sin(37°) c. 2.5 tan(83°) d. sin(45°) 220 2 cos(23°) 12.8 18.7 12. a. b. c. d. cos(14°) 5 sin(18°) tan(60°32′) sin(35°25′) 55.7 3.8 tan(1°51′) 2.5 sin(27°8′) 3.2 cos(34°52′) 13. a. b. c. d. cos(89°21′) 4.5 sin(25°45′) 10.4 cos(83°2′) 0.8 sin(12°48′) 460 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition 14. For parts a to f, write an expression for: i. sine ii. cosine iii. tangent. a. d b. c. θ i h β k α j e g f l d. e. f. o a b γ n β c u m v γ t Understanding 15. WE10 Write the equation that relates the two marked sides and the marked angle in each of the following triangles. a. b. c. 9 22 θ θ 18 15 7 30 θ 16. Write the equation that relates the two marked sides and the marked angle in each of the following triangles. a. b. c. 18.6 3.6 p 13 t θ 25° 23.5 α Communicating, reasoning and problem solving 17. Consider the right-angled triangle shown. α 37° a. Label each of the sides using the letters O, A and H with respect to the 37° angle. b. Determine the value of each trigonometric ratio. (Where applicable, answers should be given correct to 2 decimal places.) i. sin(37°) ii. cos(37°) iii. tan(37°) TOPIC 9 Trigonometry I 461 18. Consider the right-angled triangle shown in question 17. a. Determine the value of each of these trigonometric ratios, correct to 2 decimal places. (Hint: First relabel the sides of the triangle with respect to angle 𝛼) i. sin(𝛼) ii. cos(𝛼) iii. tan(𝛼) b. What do you notice about the relationship between sin(37°) and cos(𝛼)? Explain your answer. c. What do you notice about the relationship between sin(𝛼) and cos(37°)? Explain your answer. d. Make a general statement about the two angles. 19. Using a triangle labelled with a, h and o, algebraically show that tan(𝜃) = sin(𝜃). cos(𝜃) (Hint: Write all the sides in terms of the hypotenuse.) 20. ABC is a scalene triangle with side lengths a, b and c as shown. Angles BDA and BDC are right angles. B c a h b–x x A D C b a. Express h2 in terms of a and x. Equate the two equations for h2 to show that c2 = a2 + b2 − 2bx. b. Express h2 in terms of b, c and x. Use your knowledge of trigonometry to produce the equation c2 = a2 + b2 − 2ab cos(C), which is known c. d. as the cosine rule for non-right-angled triangles. 21. Determine the length of the side DC in terms of x, y and 𝜃. B x A y θ C D 22. Explain how we determine whether to use sin, cos or tan in trigonometry questions. cliff to the top of the cliff is 𝛼°. There is a lighthouse with height t on the cliff. From the observer, the angle 23. From an observer on a boat 110 m away from a vertical cliff with height c, the angle from the base of the from the base of the lighthouse to the top of the lighthouse is another 𝜃° more than 𝛼°. Express the height of the lighthouse, t, in terms of 𝜃° and 𝛼°. t θ c α 110 m 462 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition  LESSON 9.5 Using trigonometry to calculate side lengths LEARNING INTENTION At the end of this lesson you should be able to: select and use appropriate trigonometric ratios to find the length of an unknown side when the length of one other side and an acute angle is known. 9.5.1 Using trigonometry to calculate side lengths eles-4804 When one acute angle and one side length are known in a right-angled triangle, this information can be used to find all other unknown sides or angles. WORKED EXAMPLE 11 Using trigonometry to calculate side lengths Calculate the value of each pronumeral, giving answers correct to 3 decimal places. a. b. 6 cm a 32° 0.346 cm f 35° THINK WRITE/DRAW a. 1. Label the marked sides of the triangle. a. H O 6 cm a 35° sin(𝜃) = O 2. Identify the appropriate trigonometric ratio to use. H 3. Substitute O = a, H = 6 and 𝜃 = 35°. sin(35°) = a 6 6 sin(35°) = a a = 6 sin(35°) 4. Make a the subject of the equation. 5. Calculate and round the answer, correct to a ≈ 3.441 cm 3 decimal places. b. 1. Label the marked sides of the triangle. b. 32° H A 0.346 cm f TOPIC 9 Trigonometry I 463  cos(𝜃) = A 2. Identify the appropriate trigonometric ratio to use. H 3. Substitute A = f, H = 0.346 and 𝜃 = 32°. cos(32°) = f 0.346 0.346 cos(32°) = f f = 0.346 cos(32°) 4. Make f the subject of the equation. 5. Calculate and round the answer, correct to f ≈ 0.293 cm 3 decimal places. WORKED EXAMPLE 12 Using trigonometry to calculate side lengths Calculate the value of the pronumeral in the triangle shown. Give the answer correct to 2 decimal places. 120 m 5° p THINK WRITE/DRAW 1. Label the marked sides of the triangle. H O 120 m 5° A p tan(𝜃) = O 2. Identify the appropriate trigonometric ratio to use. A 3. Substitute O = 120, A = p and 𝜃 = 5°. tan(5°) = 120 p 4. Make p the subject of the equation. p × tan(5°) = 120 p= i. Multiply both sides of the equation by p. 120 ii. Divide both sides of the equation by tan(5°). tan 5° 5. Calculate and round the answer, correct to p ≈ 1371.61 m 2 decimal places. DISCUSSION How does solving a trigonometric equation differ when we are finding the length of the hypotenuse side compared to when finding the length of a shorter side? Resources Resourceseses Interactivity Using trigonometry to calculate side lengths (int-6133) 464 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition Exercise 9.5 Using trigonometry to calculate side lengths 9.5 Quick quiz 9.5 Exercise Individual pathways PRACTISE CONSOLIDATE MASTER 1, 3, 7, 9, 12 2, 5, 8, 10, 13 4, 6, 11, 14, 15 Fluency 1. WE11 Calculate the value of each pronumeral in each of the following, correct to 3 decimal places. a. b. c. 8 x 10 cm a 25° a 31° 14 60° 2. WE12 Calculate the value of each pronumeral in each of the following, correct to 2 decimal places. a. b. 4.6 m c. 71° 13° 94 mm m n 68° 2.3 m t 3. Determine the length of the unknown side in each of the following, correct to 2 decimal places. a. b. p c. 14 m 8°5 2 ' 11.7 m 43.95 m t 1 2' 18° 40°26' x 4. Determine the length of the unknown side in each of the following, correct to 2 decimal places. a. b. c. 6°25' x x 80.9 cm x 75.23 km 21°25' 2' 4°4 11.2 mm 3 TOPIC 9 Trigonometry I 465  5. Calculate the value of the pronumeral in each of the following, correct to 2 decimal places. a. x b. 23.7 m c. 34 36°42' z °1 2' 12.3 m y 43.9 cm 46° 6. Calculate the value of the pronumeral in each of the following, correct to 2 decimal places. a. b. c. q 0.732 km p p 15.3 m a 73°5' 63°11' 13°12' 47.385 km b Understanding 7. Given that the angle 𝜃 is 42° and the length of the hypotenuse is 8.95 m in a right-angled triangle, calculate the length of: a. the opposite side b. the adjacent side. Give each answer correct to 1 decimal place. 8. A ladder rests against a wall. If the angle between the ladder and the ground is 35° and the foot of the ladder is 1.5 m from the wall, calculate how high up the wall the ladder reaches. Write your answer in metres correct to 2 decimal places. Communicating, reasoning and problem solving 9. Tran is going to construct an enclosed rectangular desktop that is at an incline of 15°. The diagonal length of the desktop is 50 cm. At the high end, the desktop, including top, bottom and sides, will be raised 8 cm. The desktop will be made of wood. The diagram below represents this information. Side view of the desktop Top view of the desktop z 50 cm x 8 cm 15° y a. Determine the values (in centimetres) of x, y and z of the desktop. Write your answers correct to 2 decimal places. b. Using your answer from part a determine the minimum area of wood, in cm2 , Tran needs to construct his desktop including top, bottom and sides. Write your answer correct to 2 decimal places. 10. a. In a right-angled triangle, under what circumstances will the opposite side and the adjacent side have the b. In a right-angled triangle, for what values of 𝜃 (the reference angle) will the adjacent side be longer than same length? the opposite side? 466 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition 11. In triangle ABC shown, the length of x correct to 2 decimal places is 8.41 cm. AM is perpendicular to BC. Jack found the length of x to be 5.11 cm. Below is his working. Identify his error and what he should have done instead. cos(36°) = A BM BM = 3.641 4.5 cm 4.5 sin(36°) = 36° 29° C AM B M AM = 2.645 4.5 x tan(29°) = 2.645 MC = 1.466 MC BC = 3.641 + 1.466 = 5.107 cm = 5.11 cm (rounded to 2 decimal places) 12. A surveyor needs to determine the height of a building. She measures the angle of elevation of the top of the building from two points, 64 m apart. The surveyor’s eye level is 195 cm above h the ground. 47°48 36°2 a. Determine the expressions for the height of the building, h, in ʹ 4ʹ terms of x using the two angles. x 64 m 195 cm b. Solve for x by equating the two expressions obtained in part a. Give your answer to 2 decimal places. c. Determine the height of the building correct to 2 decimal places. 13. Building A and Building B are 110 m apart. From the base of Building A to the top of Building B, the angle is 15°. From the top of Building A looking down to the top of Building B, the angle is 22°. Evaluate the heights of each of the two buildings correct to 1 decimal place. 22º Building A Building B 15º 110 m 14. PATH If angles QNM, QNP and MNP are right angles, determine the length of NQ. Q h N x y 30° 45° M 120 P 15. Determine how solving a trigonometric equation differs when we are calculating the length of the hypotenuse side compared to when determining the length of a shorter side. TOPIC 9 Trigonometry I 467  LESSON 9.6 Using trigonometry to calculate angle size LEARNING INTENTION At the end of this lesson you should be able to: select and use appropriate trigonometric ratios to find unknown angles in right-angled triangles apply inverse operations to calculate the acute angle when two sides are given. 9.6.1 Using trigonometry to calculate angle size eles-4805 The size of any angle in a right-angled triangle can be found if the lengths of any two sides are known. Just as inverse operations are used to solve equations, inverse trigonometric ratios are used to solve −1 trigonometric equations for the value of the angle. Inverse cosine (cos−1 ) is the inverse of cosine. Inverse sine (sin ) is the inverse of sine. Inverse tangent (tan−1 ) is the inverse of tangent. Inverse operations If sin(𝜃) = a, then sin−1 (a) = 𝜃. If cos(𝜃) = a, then cos−1 (a) = 𝜃. If tan(𝜃) = a, then tan−1 (a) = 𝜃. For example, since sin(30°) = 0.5, then sin−1 (0.5) = 30°; this is read as ‘inverse sine of 0.5 is 30 degrees’. A calculator can be used to calculate the values of inverse trigonometric ratios. WORKED EXAMPLE 13 Evaluating angles using inverse trigonometric ratios For each of the following, calculate the size of the angle, 𝜃, correct to the nearest degree. a. b. 5 cm 5m 3.5 cm 𝜃 θ 11 m THINK WRITE/DRAW a. 1. Label the given sides of the triangle. a. H O 5 cm 3.5 cm 𝜃 sin(𝜃) = O 2. Identify the appropriate trigonometric ratio to use. We are given O and H. H 468 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition  3. Substitute O = 3.5 and H = 5 and evaluate sin(𝜃) = 3.5 = 0.7 the expression. 5 4. Make 𝜃 the subject of the equation using 𝜃 = sin−1 (0.7) inverse sine. = 44.427 004° 5. Evaluate 𝜃 and round the answer, correct to 𝜃 ≈ 44° the nearest degree. b. 1. Label the given sides of the triangle. b. O 5m 𝜃 11 m A tan(𝜃) = O 2. Identify the appropriate trigonometric ratio to use. Given O and A. A 3. Substitute O = 5 and A = 11. tan(𝜃) = 5 11 4. Make 𝜃 the subject of the equation using 𝜃 = tan −1 5 ( ) = 24.443 954 78° inverse tangent. 11 5. Evaluate 𝜃 and round the answer, correct to 𝜃 ≈ 24° the nearest degree. WORKED EXAMPLE 14 Evaluating angles in degrees and minutes Calculate the size of angle 𝜃, correct to the nearest minute. 3.1 m 𝜃 7.2 m THINK WRITE/DRAW 1. Label the given sides of the triangle. 3.1 m A θ O 7.2 m tan(𝜃) = O 2. Identify the appropriate trigonometric ratio to use. A TOPIC 9 Trigonometry I 469  3. Substitute O = 7.2 and A = 3.1. tan(𝜃) = 7.2 3.1 4. Make 𝜃 the subject of the equation using 𝜃 = tan−1 17.2 ( ) inverse tangent. 3.1 5. Evaluate 𝜃 and write the calculator display. 𝜃 = 66.70543675° 6. Use the calculator to convert the answer to 𝜃 = 66°42′19.572′′ degrees, minutes and seconds. 7. Round the answer to the nearest minute. 𝜃 ≈ 66°42′ Resources Resourceseses Interactivity Finding the angle when two sides are known (int-6046) Exercise 9.6 Using trigonometry to calculate angle size 9.6 Quick quiz 9.6 Exercise Individual pathways PRACTISE CONSOLIDATE MASTER 1, 4, 7, 9, 12 2, 5, 8, 10, 13 3, 6, 11, 14 Fluency 1. WE13 Calculate the size of the angle, 𝜃, in each of the following. Give your answer correct to the nearest degree. a. b. c. 5.2 4.7 4.8 8 θ θ θ 3.2 3 2. WE14 Calculate the size of the angle marked with the pronumeral in each of the following. Give your answer correct to the nearest minute. a. b. 7.2 m c. β 12 17 4m θ 10 θ 12 470 Jacaranda Maths Quest 10 Stage 5 NSW Syllabus Third Edition 3. Calculate the size of the angle marked with the pronumeral in each of the following. Give your answer correct to the nearest minute.