This simple one is based on looking at various sized right angled triangles with angles 37 (36á9 ), 53 (53á1 ) and 90.

Transcription

1 TRIGONOMETRY IN A RIGHT ANGLED TRIANGLE There are various ways of introducing Trigonometry, including the use of computers, videos and graphics calculators. This simple one is based on looking at various sized right angled triangles with angles 37 (36á9 ), 53 (53á1 ) and 90. Define the terms ÔoppositeÕ, ÔhypotenuseÕ and ÔadjacentÕ. in terms of angle P. Draw 3-4 similar right angled triangles or pre-prepare a worksheet with the triangles already drawn P hyp adj opp 37 4 cm 3 cm 4á5 cm 6 cm 37 6 cm 37 8 cm Show that, in each triangle, (with 37 at angle P), the ratio defined by opp 3 4á5 6 = = = = 0á75 (is fixed) adj Define this as the tangent of angle P and write it as: tangent (37 ) = 0á75 or for short tan 37 = 0á75 Show that irrespective of the size of the triangle, as long as angle P = 37, the ratio of opp / adj will always be 0á75. Introduce the scientific calculator and have students check that tan 37 = 0á75 (0á ) Have students look up several tangents and try to get them to round answers (say, 3 d.p.) Exercise 1, question 1, may now be attempted. (5-10 minutes) Now students should be shown how to calculate the missing side in a right angled triangle by means of 3-4 examples 2á1 cm x cm x cm x cm cm 10 cm x x x tan 34 = tan 41 = tan 58 = 6 => x = 6 x tan 34 => x = 4á05 cm 10 => x = 10 x tan 41 => x = 8á69 cm 2á1 => x = 2á1 x tan 58 => x = 3á36 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Staff Notes 10

2 Exercise 1, questions 2 and 3, may now be attempted. Notes for Exercise 2 Using tangents in reverse Students should be shown how to use the Shift tan, 2nd tan, Inv tan or tan -1 buttons to find the size of angle x, given the value of its ratio. => tan = 0á515 => x = 27á2 (to 1 d.p.) => tan = 2á435 => x = 67á7 (to 1 d.p.) Exercise 2, question 1, may now be attempted. Now, students should be shown how to determine the missing side in a right angled triangle given the 2 sides. 5 cm 4 tan = 5 => tan = 0á8 => x = 38á7 4 cm 3 cm 7 cm 3 tan = 7 => tan = 0á429 => x = 23á2 3á2 cm 3á2 tan = 1á9 => tan = 1á684 => x = 59á3 1á9 cm Exercise 2, questions 2 and 3, may now be attempted. Notes for Exercises 3 to 6 The sine and cosine function. The sine and cosine functions can be introduced using similar methods to those used in the tangents function. (sine ratio is covered in Exercises 3 and 4 and cosine ratio in Exercises 5 and 6) Exercises 3 to 6 may now be attempted after the appropriate introduction. Notes for Exercise 7 In this exercise students have to decide on the appropriate ratio to be used in a question. The mnemonic SOH CAH TOA could be introduced with 3 or 4 examples to indicate its use. Mathematics Support Materials: Mathematics 3 (Int 1) Ð Staff Notes 11

3 8 cm 29 x sin 29 = 8 => x = 8 x sin 29 => x = 3á88 cm x cm 15 cm 13 cm 13 cos = 15 => cos = 0á867 => x = 29á9 x cm 6á8 cm 59 x tan 59 = 6á8 => x = 6á8 x tan 59 => x = 11á3 cm Exercise 7 may now be attempted. Then the Checkup Exercise for Trigonometry in a Right Angled Triangle. STANDARD FORM (Scientific Notation) Notes for Exercise 1 Powers (indices) The idea of taking a power of a number should be introduced or revised. 4 3 = 4 x 4 x 4 = 64 ; 5 2 = 5 x 5 = 25 ; 3 4 = 3 x 3 x 3 x 3 = 81 (in particular, emphasise the powers of 10) 10 4 = 10 x 10 x 10 x 10 = = ( a 1, followed by 4 zeros) 10 6 = 10 x 10 x 10 x 10 x 10 x 10 = = ( a 1, followed by 6 zeros) Negative powers of 10 can be defined using these examples: Ð3 = 10 3 = = 10 x 10 x = 0á Ð2 = 10 2 = = 10 x = 0á01 Students should also be shown how to use the x y button on the calculator to find, for example, Exercise 1 may now be attempted. Notes for Exercise 2 Interpret numbers expressed in Standard Form (or Scientific Notation) Examples which could be used: 3á27 x 10 3 = 3á27 x (10 x 10 x 10) = 3270 (using a calculator) 4á8 x 10 6 = 4á8 x (10 x 10 x 10 x 10 x 10 x 10) = (using a calculator) Mathematics Support Materials: Mathematics 3 (Int 1) Ð Staff Notes 12 58

4 TRIGONOMETRY IN RIGHT ANGLED TRIANGLES The Tangent Ratio Exercise 1 tan x = Opp Adj (Hyp)otenuse x (Adj)acent (Opp)osite 1. Use your calculator to look up the following (to 2 decimal places): (a) tan 37 (b) tan 51 (c) tan 70 (d) tan 61 (e) tan 78 (f) tan 17 (g) tan 43 (h) tan 39 (i) tan 89 (j) tan 27 (k) tan 69 (l) tan 58 (m) tan 25á6 (n) tan 48á7 (o) tan 6á5 2. Set down each part of this question in the same way as shown in the example opposite. Find the values of a, b, c, cm x cm x tan 32 = 6 => x = 6 x tan 32 => x = 3á75 cm (a) (b) (c) 67 8 cm a cm 37 b cm (d) (e) (f) 75 7 cm c cm d cm cm e cm 77 4á9 cm f mm 29 10á5 mm 10 cm h cm (g) 56 (h) (i) g cm 9á1 cm cm 31 i cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 20

5 3. (a) A telegraph pole is strengthened by a steel wire as shown in the diagram opposite. Calculate the height (h) of the pole. 42 7á6 m h m (b) h m 35 When a boat is 250 metres from the base of a cliff, the sailor has to raise his eyes by 35 from the horizontal in order to see the top of the cliff. Calculate the height (h) of the cliff. 250 m (c) At a point, 8á5 metres from the base of a tree, the Òangle of elevationó of the top of the tree is 29. Calculate the height (h) of the tree. h m 29 8á5 m (d) h m 32 7á5 m When the sun shines on a wall, it casts a shadow 7á5 metres long. The angle of elevation of the top of the wall from the end of the shadow is 32. Calculate the height (h) of the wall. Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 21

6 Using Tangents in Reverse Exercise 2 1. Use Shift tan, 2nd tan, Inv tan or tan -1 buttons to find the size of angle x, given: (a) tan x = 0á123 (b) tan x = 0á342 (c) tan x = 0á9 (d) tan x = 1 (e) tan x = 1á732 (f) tan x = 2 (g) tan x = 3á157 (h) tan x = 5á095 (i) tan x = 4á5 (j) tan x = 10á987 (k) tan x = 16 (l) tan x = 19á Set down each part of this question in the same way as shown in the example opposite. Find the size of angle x each time. 6 cm 3á9 cm 3á9 tan = 6 => tan = 0á515 => x = 27á2 (a) (b) (c) 6 cm 9á6 cm 8 cm 13 cm 11 cm (d) (e) (f) 7 cm 10 cm 4 mm 10 mm 7á1cm 4á9 cm 6á9 cm (g) (h) 8 cm (i) 15 cm 9á1 cm 9 cm 7 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 22

7 3 (a) A flagpole is 7á5 metres high. A strengthening wire is attached from the top of the pole to a point 6 metres from the base of the pole. Calculate the size of the angle between the wire and the ground. 7á5 m 6á0 m (b) 15 cm 25 cm The picture shows a shelf support bracket in the shape of a right angled triangle. Calculate the size of the angle marked. (c) A ladder is resting against a wall. The base of the ladder is 3á1 metres from the foot of the wall. The ladder reaches a point 5á8 metres up the wall. Calculate the size of the angle the ladder makes with the ground. 5á8 m (d) Shown below are two newly designed wheelchair ramps. 3á1 m 6á5 m STYLE 1 0á9 m y 8á7 m STYLE 2 1á3 m (i) Calculate the sizes of the angles marked and y. (ii) Which of ramp is steeper? Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 23

8 The Sine Ratio Exercise 3 sin x = Opp Hyp (Hyp)otenuse x (Adj)acent (Opp)osite 1. Use your calculator to look up the following: (a) sin 37 (b) sin 51 (c) sin 70 (d) sin 61 (e) sin 78 (f) sin 17 (g) sin 43 (h) sin 39 (i) sin 86 (j) sin 47 (k) sin 69 (l) sin 58 (m) sin 25á6 (n) sin 48á7 (o) sin 6á5 2. Set down each part of this question in the same way as shown in the example opposite. Find the values of a, b, c,... 8 cm 36 x cm x sin 36 = 8 => x = 8 x sin 36 => x = 4á70 cm (a) (b) (c) a cm b cm 10 cm 33 c cm 9 cm (d) (e) (f) 19 cm d cm 31 cm e cm f cm 60 2á3 cm (g) (h) h cm (i) i cm 6á9 cm 35 g cm cm 33 0á7 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 24

9 3. (a) A boy flies a kite on the end of a piece of string 60 metres long. The string makes an angle of 43 with the ground. Calculate the height (h) metres of the kite above ground. 60 m h m 43 (b) h m 6á5 m A slide in the park is 6á5 metres long. It makes an angle of 32 with the ground. Calculate the height (h) of the slide. 32 (c) A gate has a strengthening bar across its diagonal. The bar is 2á3 metres long, as shown in the diagram. Calculate the height (h) of the gate. 29 2á3 m h m (d) A car ramp is 4á8 metres long. It makes an angle of 15 with the ground. Calculate the height (h) of the top of the ramp above the ground. h m 4á8 m 15 (e) An isosceles triangle has each of its two sloping sides 12 centimetres long. The sides make an angle of 65 with the base. Calculate the height (h) of the triangle. 65 h cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 25

10 Using Sines in Reverse Exercise 4 1. Use Shift sin, 2nd sin, Inv sin or sin -1 buttons to find the size of angle x, given: (a) sin x = 0á223 (b) sin x = 0á376 (c) sin x = 0á9 (d) sin x = 0á853 (e) sin x = 0á732 (f) sin x = 0á5 (g) sin x = 0á866 (h) sin x = 0á195 (i) sin x = 0á445 (j) sin x = 0á987 (k) sin x = 0á333 (l) sin x = 1á1 (?) 2. Set down each part of this question in the same way as shown in the example opposite. Find the size of angle x each time. 5á2 cm 2á9 cm 2á9 sin = 5á2 => sin = 0á558 => x = 33á9 (a) (b) (c) 10 cm 8 cm 5 mm 10 mm 9 cm (d) (e) (f) 20 cm 18 cm 8á2 cm 10á3 cm 2 cm 2á9 cm 5á9 cm (g) (h) (i) 1á1 cm 6á8 cm 0á6 cm 13 cm 11 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 26

11 3 (a) A large wooden beam, 8á3 m long, is used to shore up a crumbling wall as shown. Calculate the size of the angle between the support and the ground. 8á3 m 7á2 m (b) 1á8 m 12á5 m The picture shows a roller skate ramp built in the local park. Calculate the size of the angle marked. (c) Part of a bicycle frame is in the shape of a right angled triangle. Calculate the size of the angle marked in the diagram. 46 cm 60 cm (d) The drawbridge of a castle is 9á2 metres long. In the figure, the chain is attached to the tip of the drawbridge and is horizontal. If the chain is 3á4 metres long, calculate the size of the angle between the drawbridge and the wall. 3á4 m 9á2 m x (e) 15 cm 13 cm 15 cm An isosceles triangle has its two sloping sides each 15 centimetres long. The height of the triangle is 13 centimetres. Calculate the size of the angle x between one of the long sides and the base. Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 27

12 The Cosine Ratio Exercise 5 cos x = Adj Hyp (Hyp)otenuse x (Adj)acent (Opp)osite 1. Use your calculator to look up the following: (a) cos 57 (b) cos 61 (c) cos 50 (d) cos 81 (e) cos 18 (f) cos 27 (g) cos 33 (h) cos 75 (i) cos 88 (j) cos 26 (k) cos 9 (l) cos 60 (m) cos 35á6 (n) cos 58á3 (o) cos 7á5 2. Set each part of this question down in the same way as shown in the example opposite. Find the values of a, b, c,... 6 cm 32 x cm x cos 32 = 6 => x = 6 x cos 32 => x = 5á09 cm (a) (b) (c) a cm 9 cm cm 22 cm 72 b cm (d) (e) (f) 13 cm 8 cm 55 d cm 61 e cm 69 c cm f cm 58 4á3 cm (g) (h) (i) g cm h cm 130 cm 29 0á9 cm 38 8á1 cm 21 i cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 28

13 3. (a) A farmerõs fencepost needs a support. He attaches a 2 metre wire to it and the wire makes an angle of 34 to the ground. Calculate how far the foot of the wire is from the base of the fencepost m x m (b) 17 7á2 cm x cm A door wedge has its sloping side 7á2 centimetres long. The sloping side makes an angle of 17 with the base. Calculate the length (x) of the base. (c) The diagonal of rectangle ABCD is 17 centimetres long. The diagonal AC makes an angle of 31 to the base line DC. Calculate the length (x) of the rectangle. A D 17 cm x cm 31 B C C (d) A D 26 cm 47 B A piece of wire ACB is used to hang a picture. Triangle ABC is isosceles. Triangle BCD is right angled. Calculate the length of the line BD and then write the width AB of the picture. 2á8 m (e) A diagram shows the side view of a garden shed. Calculate the width (w) of the shed. w m 22 Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 29

14 Using Cosines in Reverse Exercise 6 1. Use Shift cos, 2nd cos, Inv cos or cos -1 buttons to find the size of angle x, given: (a) cos x = 0á423 (b) cos x = 0á396 (c) cos x = 0á6 (d) cos x = 0á573 (e) cos x = 0á718 (f) cos x = 0á9 (g) cos x = 0á866 (h) cos x = 0á225 (i) cos x = 0á071 (j) cos x = 0á999 (k) cos x = 0á301 (l) cos x = 1á2 (?) 2. Set down each part of this question in the same way as shown in the example opposite. Find the size of angle x each time. 5á8 cm 4á9 cm 4á9 cos = 5á8 => cos = 0á845 => x = 32á3 (a) (b) (c) 9 cm 11 cm 7 cm 8 cm 12 mm 6 mm 1á4 cm (d) (e) (f) 3á2 cm 4 cm 9á3 cm 30 cm 13 cm (g) (h) (i) 2á2 cm 0á9 cm 1á2 cm 1á6 cm 37 cm 45 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 30

15 3 (a) A plank of wood is 6á5 metres long. It rests against the top of a wall and its foot is 5á2 metres from the base of the wall. Calculate the size of the angle between the plank and the ground. 6á5 m 5á2 m (b) TENT 2á3 m The guy rope of a tent is 2á3 metres long. It is attached to the ground at a point 1á9 metres from the foot of the tent wall. Calculate the size of the angle marked. 1á9 m (c) A balloon is 90 metres above ground. The wire which tethers it to the ground is 135 metres long. Calculate the size of the angle marked in the diagram. 135 m 90 m (d) A kite is made up of two identical right angled triangles. 47 cm Calculate the size of the angle marked in the figure. 36 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 31

16 A Mixture of Tangents, Sines and Cosines tan x = sin x = cos x = Opp Adj Opp Hyp Adj Hyp (Hyp)otenuse x (Adj)acent (Opp)osite Exercise 7 SOH CAH TOA 1. Find the values of a, b, c,... (a) (b) (c) a cm b cm cm 7 cm 71 c cm (d) (e) (f) d cm e cm 11á5 mm á9 cm 19 f mm g cm h cm (g) 58 (h) (i) 11á1 cm cm 36 i cm 10 cm 6á7 cm (j) j (k) (l) 11 cm 9á1 cm k 15 cm l contõd... Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 32

17 (m) (n) (o) 9á2 cm m 8á4 cm n 6á9 cm 10á5 mm o 9á5 mm 2. (a) A plastic set-square has one of its angles 60. The longest side is 11á5 centimetres long. Calculate the size of the shortest side marked x cm. 11á5 cm 60 x cm (b) 15á3 m A radio mast has a support wire attached from the ground to a point 15á3 metres up the mast. Calculate the size of the angle between the wire and the ground. 18á2 m (c) A mine shaft slopes at an angle of 15 with the ground. The shaft is 220 metres long. Calculate the distance d metres from the mine entrance to the top of a vertical air-shaft which comes up from the foot of the mine m d m (d) Shown is a wooden roof support. The vertical strip of wood cuts the shape into 2 right angled triangles. (i) Calculate the height (h )metres of the vertical strip. (ii) Calculate the size of the angle marked m h m 14 m Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 33

18 Checkup for Trigonometry in a Right Angled Triangle 1. Calculate the length of the sides marked x, y and z. (a) (b) (c) 58 7 cm x cm 18 cm 28 y cm 2. Calculate the length of the angles marked a, b and c. (a) (b) 7á2 cm (c) 17 cm a 11 cm 9á1 cm 3. (a) The diagram shows a plane coming in to land. Calculate the size of the angle the plane will make with the ground as it lands. b 5á7 mm 33 z cm 35 cm 10á5 mm c 380 m 1300 m (b) x m 1á8 m A long handled brush is leaning against a wall. The brush is 1á8 metres long. It makes an angle of 72 with the ground. Calculate how far up the wall the top of the brush handle reaches. C cm 30 cm (c) Triangle ABC is isosceles. Triangle ADC is right angled. AC and BC are both 30 centimetres long. AB is 20 centimetres long. Calculate the size of the angle marked. A D 20 cm Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials B 34

19 Trigonometry in a Right Angled Triangle Exercise 1 1. (a) 0á75 (b) 1á24 (c) 2á75 (d) 1á80 (e) 4á71 (f) 0á31 (g) 0á93 (h) 0á81 (i) 57á29 (j) 0á51 (k) 2á61 (l) 1á60 (m) 0á48 (n) 1á14 (o) 0á11 2. (a) 18á85 (b) 9á04 (c) 26á12 (d) 20á69 (e) 21á22 (f) 5á82 (g) 14á83 (h) 8á49 (i) 9á01 3 (a) 6á84 m (b) 175á05 m (c) 4á71 m (d) 4á69 m Exercise 2 1. (a) 7á01 (b) 18á88 (c) 41á99 (d) 45 (e) 60 (f) 63á43 (g) 72á42 (h) 78á90 (i) 77á47 (j) 84á80 (k) 86á42 (l) (a) 26á57 (b) 50á19 (c) 49á76 (d) 55á01 (e) 21á80 (f) 55á39 (g) 37á17 (h) 28á07 (i) 52á13 3. (a) 51á34 (b) 30á96 (c) 61á88 (d) (i) x = 7á88, y = 8á50 (ii) 2 Exercise 3 1. (a) 0á60 (b) 0á78 (c) 0á94 (d) 0á88 (e) 0á98 (f) 0á29 (g) 0á68 (h) 0á63 (i) 1á00 (j) 0á73 (k) 0á93 (l) 0á85 (m) 0á43 (n) 0á75 (o) 0á11 2. (a) 10á60 (b) 5á45 (c) 7á72 (d) 30á32 (e) 18á07 (f) 1á99 (g) 3á96 (h) 137á77 (i) 0á38 3. (a) 40á92 m (b) 3á44 m (c) 1á12 m (d) 1á24 m (e) 10á88 cm Exercise 4 1. (a) 12á89 (b) 22á09 (c) 64á16 (d) 58á54 (e) 47á05 (f) 30 (g) 60 (h) 11á24 (i) 26á42 (j) 80á75 (k) 19á45 (l) error 2. (a) 53á13 (b) 30 (c) 48á59 (d) 64á16 (e) 52á76 (f) 43á60 (g) 60á19 (h) 33á06 (i) 57á80 3. (a) 60á17 (b) 8á28 (c) 50á06 (d) 21á69 (e) 60á07 Exercise 5 1. (a) 0á54 (b) 0á48 (c) 0á64 (d) 0á16 (e) 0á95 (f) 0á89 (g) 0á84 (h) 0á26 (i) 0á03 (j) 0á90 (k) 0á99 (l) 0á50 (m) 0á81 (n) 0á53 (o) 0á99 Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 49

20 2. (a) 7á71 (b) 4á94 (c) 7á88 (d) 7á46 (e) 3á88 (f) 2á279 (g) 0á79 (h) 102á44 (i) 7á56 3. (a) 1á66 m (b) 6á89 cm (c) 14á57 cm (d) 35á46 cm (e) 2á60 m Exercise 6 1. (a) 64á98 (b) 66á67 (c) 53á13 (d) 55á04 (e) 44á11 (f) 25á84 (g) 30 (h) 77á0 (i) 85á93 (j) 2á56 (k) 72á48 (l) error 2. (a) 38á94 (b) 43á34 (c) 60 (d) 64á06 (e) 64á53 (f) 64á32 (g) 43á34 (h) 41á41 (i) 34á69 3. (a) 36á9 (b) 34á3 (c) 48á2 (d) 40 Exercise 7 1. (a) 16á64 (b) 6á36 (c) 2á28 (d) 10á88 (e) 11á33 (f) 10á87 (g) 6á36 (h) 7á28 (i) 18á16 (j) 47á7 (k) 47á4 (l) 36á9 (m) 50á1 (n) 50á6 (o) 25á2 2. (a) 5á75 cm (b) 40á1 (c) 212á5 m (d) (i) 7á03 m (ii) 30á2 Checkup Exercise 1. (a) 11á2 (b) 15á89 (c) 19á06 2. (a) 49á7 (b) 51á6 (c) 32á9 3. (a) 16á3 (b) 1á71 m (c) 70á5 Standard Form Exercise 1 1. (a) 100 (b) (c) (d) 10 (e) (f) (a) 0á01 (b) 0á0001 (c) 0á (d) 0á1 (e) 0á00001 (f) 0á (a) (b) (c) (d) 0á00001 (e) 0á1 (f) 1 Exercise 2 1. (a) (b) (c) 315 (d) (e) (f) (g) 72á6 (h) (a) 0á064 (b) 0á00019 (c) 0á (d) 0á00161 (e) 0á0493 (f) 0á (g) 0á705 (h) 0á (a) 431 (b) (c) 98á1 (d) 2760 (e) (f) (g) (h) 0á092 (i) 0á (j) 0á00798 (k) 0á (l) 0á801 Mathematics Support Materials: Mathematics 3 (Int 1) Ð Student Materials 50