Mathematics Trigonemetry

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Right-angled triangles

Example Find the length of the unknown side x in the triangle shown. Answer correct to two decimal places.

Example A vertical tent pole is supported by a rope tied to the top of the pole and to a peg on the ground. The rope is 3m in length and makes an angle of 29º to the horizontal. What is the height of the tent pole? Answer correct to two decimal places.

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 1 Find the length of the unknown side x in each triangle, correct to two decimal places

Question 2 Find the unknown angle Ө in each triangle. Answer correct to the nearest minute

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 3 Find the length of the unknown side x in each triangle, correct to two decimal places

Question 4 Find the unknown angle Ө in each triangle. Answer correct to the nearest degree

Question 5 Jack measures the angle of elevation to the top of a tree from a point on level ground as 35º. What is the height of the tree if Jack is 50m from the base of the tree? Answer to the nearest metre.

Question 6 A 3m ladder has its foot 1.5m out from the base of the wall. What angle does the ladder make with the ground? Answer correct to the nearest degree.

Bearings Page | 3 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

There are two types of bearings: 1) Compass bearings 2) True bearings

Compass Bearings Example A boat leaves Sydney and travels 420 km on a bearing of N37ºE. How far did the boat travel due north, to the nearest kilometre?

True Bearings Example Joel walks north for 3.2 km and then west for 4.4 km. What is his true bearing from his starting point, to the degree?

Question 1

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Aaron runs a distance of 7.2 km in the SE direction. How far east has Aaron run? Answer correct to one decimal place.

Question 2 A plane is travelling on a bearing of 030º from A to B. a) What is the compass bearing of A to B? b) What is the true bearing of B to A? c) What is the compass bearing of B to A?

Question 3 The diagram shows the position of P, Q and R relative to S. In the diagram, R is NE of S, Q is NW of S and ∠PSR is 155ᵒ. a) What is the true bearing of R from S? b) What is the true bearing of Q from S? c) What is the true bearing of P from S?

Question 4 The bearing of E from D is N38ºE, F is east of D and ∠DEF is 87ᵒ a) Find the values of x and y b) What is the compass bearing of E from F? c) What is the true bearing of E from F?

Question 5 Mia cycled for 15 km west and then 24 km south a) What is the value of Ө to the nearest degree? b) What is Mia’s true bearing from her starting point? c) What is Mia’s compass bearing from her starting point?

Question 6 A boat sails 137 k from Port Stephens on a bearing on 065ºT a) How far east has the boat sailed? Answer correct to one decimal place b) How far north has the boat sailed? Answer correct to one decimal place Question 7 A ship sails 5 kilometres west, then 5 kilometres south Page | 5 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

a) What is the compass bearing of the ship from its original position? b) What is the true bearing of the ship from its original position?

Question 8 Harry travelled for 8.5 km on a bearing of S30ºW from his home a) How far west is Harry from home? Answer correct to two decimal places b) How far south if Harry from home? Answer correct to two decimal places c) What is the compass bearing of his home from his current position?

Question 9 A plane left from O and travelled 350 km in the direction 225ºT to reach P. It then changed direction and travelled due north for 500 km to reach point N a) What was the distance from P to M? (Answer correct to two decimal places) b) What was the distance from M to O? (Answer correct to two decimal places) c) What was the distance from M to N? (Answer correct to two decimal places) d) What is the angle Ө, correct to the nearest degree? e) What is the true bearing of N from O?

[Trigonometry with obtuse angles] The Sine Rule

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Example Find the value of x correct to one decimal place

Example Hannah is standing 4.5m from the base of a 3m sloping wall. The angle of elevation to the top of the wall is 36º. Find the angle Ө at the top of the wall, to the nearest minute.

Question 1 Find the length of the unknown side x in each triangle, correct to two decimal places

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 2 Find the length of the unknown side x in each triangle, correct to two decimal places

Question 3 Benjamin is planning to build a triangular garden for his daughter. The vertices of the triangle are named PQR. He measured PQ as 2.7m, OR as 3.1 m and ∠PRQ as 57ᵒ. Use the sine rule to find the size of ∠RPQ, correct to the nearest degree.

Question 4 Triangle ABC has ∠ABC = 71ᵒ, ∠BCA = 69ᵒ and ∠CAB = 40ᵒ. The length of AB is 10 a) What is the value of a, correct to one decimal place? b) What is the value of b, correct to one decimal place?

Question 5 Refer to the picture on the right. Use the sine rule to:

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

a) Find the size of angle XYZ. Give your answer to the nearest degree b) Find the size of y. Give your answer to the nearest centimetre

Question 6 Sienna was located at X and saw a fire in the direction N15ºE. Seven kilometres to the east of X at Z, Dylan saw the fire in the direction N50ºW a) How far is X from the fire? Answer in kilometres correct to one decimal place b) How far is Z from the fire? Answer in kilometres correct to one decimal place

Question 7 Chelsea is travelling due east from A to B. Unfortunately, the road is blocked and she makes a detour by traveling from A to C a distance of 30 km, on a bearing of 040º. Chelsea then turns and travels south-east until she reaches B. a) What are the sizes of ∠CAB and ∠CBA? b) How far did Chelsea travel from C to B? Answer correct to the nearest kilometre c) What was the extra distance travelled on the detour? Answer correct to the nearest kilometre

Area of a triangle

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Example Find the area of the triangle to the nearest square centimetre

Question 1 In triangle ABC, side a is 36 cm, side b is 48cm and angle C is 68º. Find the area of the triangle. Answer correct to two decimal places

Question 2 In the triangle XYZ, side x is 4m, side y is 7m and angle Z is 34º. Find the area of the triangle. Answer correct to two decimal places

Question 3 A parallelogram has sides of length 7cm and 5.2 cm, and the included angle of 130º. Calculate its area, correct to one decimal place.

Question 4 A kite has adjacent sides of length 4cm and 7cm. The longer diagonal makes angles of 45º and 23.8º with the sides of the kite. Find the area if the kite.

Question 5 Find the area of each triangle, correct to one decimal place

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 6 In triangle DEF, the length of DF is 21cm, EF is 28cm, ∠FDE is 64ᵒ and ∠DEF is 43ᵒ. Find the area of triangle DEF to the nearest square centimetre

Question 7 A parallelogram PQRS has PS = 4cm, SR = 5cm and ∠PSR = 40ᵒ a) What is the area of triangle PRS? Answer correct to one decimal place b) What is the area of the parallelogram? Answer correct to one decimal place

Question 8 A drawing of a farmer’s property is shown below a) What is the area of triangle WXZ? b) What is the area of triangle XYZ? c) Find the total area of the property in square kilometres.

The Cosine Rule

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Example Find the value of x correct to two decimal places

Example Find the value of the angle Ө. Answer in degrees, correct to one decimal place

Question 1 DEF is a triangle for which DF = 37cm, EF = 46cm and ∠DFE = 44ᵒ. Use the cosine rule to find the length of DE, to the nearest millimetre. Page | 12 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 2 A triangle RST has ∠RST = 51ᵒ, ∠STR = 63ᵒ, RT = 40 and RS = 48 a) What is the size of ∠TRS? b) Find the length of x using the cosine rule. Answer correct to three significant figures

Question 3 Ruby drives a four-wheel drive along a track from point A due west to a point B, a distance of 14 km. She then turns and travels 19 km to point C. Use the cosine rule to calculate the distance Ruby is from her starting point. Answer correct to one decimal place.

Question 4 Passengers in a car travelling east, along a road that runs west-east, see a castle 10km away in the direction N65ºE. When they have travelled a further 4 km east along the road, what will be the distance to the castle? Answer correct to two decimal places?

Question 5 A stepladder has legs of length 120cm and the angle between them is 15º. Calculate the distance (to the nearest centimetre) between the legs on the ground.

Question 6 A triangle has sides measuring 4m, 5m and 7m. a) What is the size of the smallest angle in this triangle? b) What is the size of the largest angle in this triangle?

Question 7 A running circuit is in the shape of a triangle with lengths of 6 km, 6.5 km and 7 km. What are the sizes of the angles (in minutes) between each of the sides? Page | 13 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 8 The lengths of the sides of triangle ABC are in the ratio 7: 8: 9. Find the size of each angle, correct to the nearest minute.

Mixed questions Question 1 The dimensions of a block of land are shown opposite a) What is the length of x, correct to the nearest metre? b) What is the length of y, correct to the nearest metre?

Question 2 Madison measures the angle of elevation to the top of a wall as 32°. She walks 10m horizontally towards the wall and measures the angle of elevation as 51°. Find the height of the wall.

Question 3 The bearing of Y from X is 240° and the distance of Y from X is 20km. a) What is the value of a? b) If Z is 18km due north of X, calculate the distance of Y from Z, correct to the nearest kilometre c) What is the value of b to the nearest degree?

Question 4 Andrew plans to build a triangular flower bed in the lower part of a circular garden. The length is TS is 24m, TSR is 45° and RTS is 40°. a) What is the size ofSRT? b) Use the sine rule to find the length of TR. Answer correct to the nearest metre.

Question 5

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

The diagram shoes information about the locations of towns P, Q and R. Amber takes 2 hours and 30 minutes to walk directly from Town P to Town Q a) What is Amber’s walking speed, correct to the nearest km/h. b) What is the distance from P to R? Answer correct to the nearest kilometre c) How long would it take Amber to walk from P to R? Answer to nearest minute

Question 6 M is 50km north of O. The bearing of N from M is 108° and from O it is 061°. Answer the following questions to the nearest kilometre a) What is the distance between M and N? b) What is the distance between N and O?

Question 7 The diagram shows five roads: DE, DG, EF, GE and GF. The bearing of FG is 304°. ED is 292° and DG is 040°. The distance from E to F is 8lm and E is due west of F. a) Find the size of GFE b) What is the distance GE? Answer correct to one decimal place c) What is the size of DGE, GED and GDE? (Answer to the nearest degree.) d) What is the distance DE? Answer correct to one decimal place e) What is the distance DG? Answer correct to one decimal place

Question 8 A survey of a park is shown in the diagram. A path is proposed from B to D and a fence is required from B to C to D. a) What is the length of the path? Answer correct to the nearest metre b) Calculate the required length of fencing. Answer correct to the nearest metre c) What is the area of the park? Answer correct to the nearest square metre. Surveying -

Offset survey Page | 15 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Example The field book entry and field diagram from an offset survey are shown opposite. Find the area of the quadrilateral ABCD. Measurements are in metres. Answer correct to nearest metre.

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Radial Survey

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Example A plane-table radial survey is shown opposite. What is the area of ABC? Answer correct to one decimal place.

Example A compass radial survey for a field is shown opposite Page | 17 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

a) What is the size of POQ? b) What is the size of POR? c) Calculate the length of PQ correct to two decimal places d) Calculate the length of PR correct to two decimal places e) What is the perimeter of the field? Answer correct to three significant figures.

Question 1 Find the area of the following fields. Units are metres Page | 18 | CLASSWORK

HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

Question 2 The diagram on the right shows a block of land that has been surveyed. All measurements are in metres a) Find the area of the quadrilateral ABCD. Answer correct to the nearest square metre b) What is the length of AB? Answer correct to the nearest metre c) What is the length of AC? Answer correct to the nearest metre

Question 3 The field book entry on the right shoes a block of land that has been surveyed. All measurements are in metres a) Find the area of the block ABCDE. Answer correct to one decimal place b) What is the length of AC? Answer correct to the nearest metre c) What is the length of CE? Answer correct to the nearest metre

Question 4

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HSC MATHEMATICS GENERAL 2

MEASUREMENT MM5: APPLICATIONS OF TRIGONOMETRY

A radial survey of land DEFG is shown opposite a) What is the size of FOG? b) Find the area of the triangle FOG to the nearest square metre c) What is the length of FG correct to the nearest metre?

Question 5 The diagram opposite is a compass radial survey of field VWXYZ. All distances are in metres. Answer the following questions correct to one decimal place a) b) c) d) e) f)

What is the length of XY? What is the length of ZY? What is the length of VZ? What is the length of VW? What is the length of XW? Calculate the perimeter of the field

Question 6 A radial survey of land ABCD is shown a) What is the size of AOB? b) What is the area of AOB? Answer correct to the nearest square metre c) What is the area of the land? Answer correct to the nearest square metre

Question 7 Alex, Blake and Connor are standing in a field. Connor © is 15 metres away from Blake (B) on a bearing of 032°. Alex (A) is 20 metres away from Blake on a bearing of 315°. a) Draw a diagram to represent the positions A, B and C. Mark the information from the question on the diagram b) What is the size of ABC to the nearest degree? c) What is the area of ABC? Answer correct to the nearest square metre d) How far is Alex from Connor? Answer correct to the nearest metre.

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