1616 16.1Applications in Two-dimensional Problems 16.2Basic Terminology in Three-dimensional Figures...

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16.1 Applications in Two-dimensional Problems

16.2 Basic Terminology in Three-dimensional Figures

16.3 Applications in Three-dimensional Problems

Chapter Summary

Case Study

Trigonometry (3)

P. 2

As shown in the figure, M and N are the mid-points of BQ and DQ respectively.

Case StudyCase Study

Although PMA is longer than PA, PM and MA are less steep than PA, and hence they are more comfortable to walk up.

It is more comfortable to walk up an inclined road along a zigzag path, let me explain it to you. How can we walk up the hill in a r

elatively more comfortable way?

In order to explain the above question, we can use a prism to illustrate the situation such that AB is the top of the inclined road and PQ is the horizontal ground level.

P. 3

When we observe an object above us, the angle between our line of sight and the horizontal is called the angle of elevation.

1166 .1 .1 Applications in Two-dimensionalApplications in Two-dimensional ProblemsProblems

When we observe an object below us, the angle between the line of sight and the horizontal is called the angle of depression.

These two angles are important in solving practical trigonometric problems.

A. Angle of Elevation and Angle of DepressionA. Angle of Elevation and Angle of Depression

P. 4

Example 16.1TIn the figure, TB is a flag. The angles of elevation from apoint A to the top T and the base B of the flag are 35 and 20 respectively. If the flag is 3 m long, find the distance between A and T.


TAB 35 20

sin15

m 3

110sin

AT

m 15sin

110sin3

AT

m 9.10 (cor. to 3 sig. fig.)

Solution:


15TBA 90 20 (ext. of )

110By sine formula,

P. 5

Example 16.2T

Solution:


8sin

m 25

6sin

BC

m 18.7767BC

m 7767.1814sin

CD

m 5425.4CD

The height of the platform (4.5425 1.7) m

m 3 (cor. to the nearest m)


Fanny looks down from a platform at one end of a swimmingpool. There is a boy A at the far end of the pool and another boy B in the pool between A and the platform. The boys are 25 m apart and in the same lane. The angles of depression of boy A and boy B from Fanny are 6 and 14 respectively. (a) Find the height of the platform if Fanny’s eyes

are 1.7 m above the platform.

(a) By sine formula,

ACB 8CAB 6 (alt. s, // lines)

CBD 14 (alt. s, // lines)

In CBD,

P. 6

Example 16.2T


m 7767.18cos14

BD BD 18.2190 m

18 m(cor. to the nearest m)

Boy B is 18 m from the near end of the pool.


Fanny looks down from a platform at one end of a swimmingpool. There is a boy A at the far end of the pool and another boy B in the pool between A and the platform. The boys are 25 m apart and in the same lane. The angles of depression of boy A and boy B from Fanny are 6 and 14 respectively. (a) Find the height of the platform if Fanny’s eyes

are 1.7 m above the platform.(b) How far is boy B from the near end of the pool? (Give the answers correct to the nearest m.)

Solution:

(b) In CBD,CBD 14 (alt. s, // lines), BC 18.7767 m

P. 7


B. BearingB. Bearing

In junior forms, we learnt how to use a compass bearing or a true bearing to indicate the direction of an object from a given point.

Compass bearing is also known as reduced bearing, and true bearing is also known as whole circle bearing.

When using a compass bearing, directions are measured from the north (N) or the south (S), thus the bearing is represented in the form:

N E, N W, S E or S W, where 0 90. Notes: If 0 or 90, we simply write it as N, E, S or W.

When using a true bearing, all directions are measured from the north in a clockwise direction.

The bearing is expressed in the form , where 0 360 and written in three digits such as 007, 056 or 198.

P. 8

Example 16.3T


m 70cos)3000)(3600(230003600 22 FE

m 364.9 572 14m3767.3817

m 4.3817 (cor. to 1 d. p.)


Eric and Frank are cycling away from P. Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is cycling in the direction 220 with a speed of 10 m/s. After five minutes, they stop and take a rest. (a) What is the distance between them now? (Give the answers correct to 1 decimal place.)

Solution:(a) PE (12 60 5) m 3600 m PF (10 60 5) m 3000 m FPE 220 150 70

By cosine formula, 3600 m3000 m

3817.3767 m

P. 9

Example 16.3T


(cor. to 1 d. p.)

P

PE

E

PF

sinsin

3767.3817

70sin3000sin

E 7385.0

6026.47E

True bearing 360 47.6026 30

4.282


Eric and Frank are cycling away from P. Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is cycling in the direction 220 with a speed of 10 m/s. After five minutes, they stop and take a rest. (a) What is the distance between them now? (b) Find the true bearing of Frank from Eric.(Give the answers correct to 1 decimal place.)

Solution:(b) a 180 150 30, b a 30 (alt. s, // lines)

By sine formula,3600 m

3000 m

3817.3767 m

P. 10

Example 16.3T


)min 9.2(or s 53.5min 2


Eric and Frank are cycling away from P. Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is cycling in the direction 220 with a speed of 10 m/s. After five minutes, they stop and take a rest. (a) What is the distance between them now? (b) Find the true bearing of Frank from Eric.(c) After having a rest, if they cycle towards each other

at the same speed as before, how long does it takefor them to meet?

(Give the answers correct to 1 decimal place.)

3600 m3000 m

3817.3767 m

Solution:

(c) Time taken s 10)(12

m 3767.3817

(cor. to 1 d. p.)

P. 11

1166 ..22 Basic Terminology in Three-Basic Terminology in Three- dimensional Figuresdimensional Figures

A. Terms and DefinitionsA. Terms and Definitions

1. Angle between Two Straight Lines

The figure shows two intersecting straight lines lying on the same plane.

The acute angle is called the angle between the two straight lines AB and CD.

In 3-D Figures, we can also identify the angle between two straight lines.

For example, the angle between BH and FH is BHF.

P. 12


Remark: If the line is perpendicular to the plane, then the projection of the line on the plane is only a point.


2. Angle between a Straight Line and a Plane

When a line is inclined on a plane, we can get a projection of the line on the plane.

For example, when a javelin TP hits the ground, the line AP is the projection of TP on the ground.

In three-dimensional space, the angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane.For example,the projection of the line AG on AEHD is AH.the angles between the line AG and AEHD is GAH.

P. 13



3. Angle between Two Planes

Consider the following two situations.

(a) A wooden door is opened.(b) A greeting card is standing on a table.

In the above two cases, we observe that there are two planes intersecting with each other.

P. 14



When two planes intersect, they meet at a straight line which is called the line of intersection.

In the figure, and are two planes while AQB is the line of intersection. PS and RT are lines on the planes and respectively such that PQ AB and RQ AB. The angle between the lines PQ and RQ is called the angle between planes and .

Remarks:1. Actually, the angle between

two intersecting planes can be acute or obtuse.

2. Usually, we do not consider the reflex angle as the angle between two intersecting planes.

P. 15



The figure shows a rectangular block.

For planes ABFE and BCHE: Line of intersection: _____________ Angle between 2 planes: _____________

BCABE / DCH

For planes CDEF and EFGH: Line of intersection: _____________ Angle between 2 planes: _____________

EFCFG / D

The figure shows a right pyramid with a square base.

For planes VCD and ABCD: Line of intersection: _____________ Angle between 2 planes: _____________

CDVPQ

For planes VBC and VCD: Line of intersection: _____________ Angle between 2 planes: _____________

VCBND

P. 16



4. Distance between a Point and a Straight Line

Consider a rectangular pyramid.

The distance between the point B and the line VC is the perpendicular distance between B and VC, that is, the length of BE.

5. Distance between a Point and a Plane

The distance between a point and a plane is the distance between the point and its projection on the plane, that is, the perpendicular distance between the point and the plane.

As shown in the figure, PQ is the distance between point P and the plane.

P. 17

Example 16.4T


(a) In EFG, 222 FGEFEG (Pyth. theorem)

cm63 22 EGcm 45

In AEG, 222 EGAEAG (Pyth. theorem)

cm)45(4 22 AG

cm 61

(b) FAG is the angle between the lines AG and AF. In AFG,

AG

FGFAG sin

61

6

FAG 50.2(cor. to 3 sig. fig.)

The angle between the lines AG and AF is 50.2.


The figure shows a cuboid. AB 3 cm, AD 6 cm and BF 4 cm. (a) Find the length of AG and express the answer in surd form. (b) Find the angle between the lines AG and AF.

(Give the answer correct to 3 significant figures.)

Solution:

P. 18


(a) In VPD, 222 PDVPVD (Pyth. theorem)

222 515 VPcm 200VPcm 210

In BCP, 222 CPBCBP (Pyth. theorem)

cm512 22 BPcm 169

cm 13


The figure shows a regular rectangular pyramid with base12 cm 10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a) Find VP and BP and give the answers in surd form if necessary. Solution:

Example 16.5T

P. 19

Example 16.5T


(b) (i) BVD is the angle between lines VB and VD. In BCD,

222 CDBCBD (Pyth. theorem)

cm 244

In BVD, by cosine formula,

)15)(15(2

)244(1515cos

222 BVD

450

206

BVD 62.8 (cor. to 3 sig. fig.)

The angle between lines VB and VD is 62.8.


The figure shows a regular rectangular pyramid with base12 cm 10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a) Find VP and BP and give the answers in surd form if necessary.(b) Find the angle between (i) lines VB and VD, (ii) lines VP and BP. (Give the answers correct to 3 significant figures.)Solution:

cm 1012 22 BD

P. 20

Example 16.5T


)210)(13(2

15)210(13cos

222 BPV

2260

144

BPV 66.9 (cor. to 3 sig. fig.)


The figure shows a regular rectangular pyramid with base12 cm 10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a) Find VP and BP and give the answers in surd form if necessary.(b) Find the angle between (i) lines VB and VD, (ii) lines VP and BP. (Give the answers correct to 3 significant figures.)Solution:

(b) (ii) BPV is the angle between lines VP and BP.In BVP, by cosine formula,

The angle between lines VP and BP is 66.9.

P. 21

Example 16.6T


222 ADABBD (Pyth. theorem)

cm 1610 22 BDcm 356

cm 9.18 (cor. to 3 sig. fig.)

In CDF,CD AB 10 cm

cm 1035sin

DF

cm 74.5DF (cor. to 3 sig. fig.)

(b) Since BF is the projection of BD on plane EBCF, DBF is the required angle.

BD

DFDBF sin

DBF 17.7 (cor. to 3 sig. fig.)


The figure shows a wedge with rectangular planes ABCD,EFDA and EBCF. AB 10 cm, BC 16 cm, DCF 35 and DFCF. (a) Find the lengths of BD and DF. (b) Find the angle between line BD and plane EBCF. (Give the answers correct to 3 significant figures.)

(a) In ABD,

The angle between line BD and plane EBCF is 17.7.

356

35sin10

Solution:

P. 22

Example 16.7T


(a) Since EC is the projection of AC on the plane BCFE, ACE is the required angle.

In CEF, 222 CFEFCE (Pyth. theorem)

cm 815 22 CE 17 cm

In ACE,

CE

AEACE tan

17

6

ACE 19.4 (cor. to 3 sig. fig.)

The angle between the line AC and the plane BCFE is 19.4.


The figure shows a right-angled triangular prism with ABCD,AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF 6 cm, FC 8 cm and AD 15 cm. Find the angle between (a) the line AC and the plane BCFE; (b) the line AP and the plane BCFE. (Give the answers correct to 3 significant figures.)

Solution:

P. 23

Example 16.7T


In BEP, BP 7.5 cm. (Pyth. theorem)

222 BPBEEP cm 5.78 22 EP

cm 25.120

In AEP,

EP

AEAPE tan

25.120

6


The figure shows a right-angled triangular prism with ABCD,AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF 6 cm, FC 8 cm and AD 15 cm. Find the angle between (a) the line AC and the plane BCFE; (b) the line AP and the plane BCFE. (Give the answers correct to 3 significant figures.)

Solution:(b) Since EP is the projection of AP on the plane BCFE, APE is th

e required angle.

APE 28.7 (cor. to 3 sig. fig.)

The angle between the line AP and the plane BCFE is 28.7.

P. 24

Example 16.8T

Solution:


(a) ABC ACB (base s, isos. )

2

90180 ( sum of )

45

In ABM,

cm 445sin

AM

cm 22AM

(b) VMA is the angle between the planes ABC and VBC.

In AVM, 22

5tan VMA VMA 60.5 (cor. to 3 sig. fig.)


The figure shows a pyramid with a right-angled triangular base. AB AC 4 cm, VA 5 cm and VAB VAC 90. (a) Find the length of AM where M is the mid-point of BC in surd form. (b) Find the angle between the planes ABC and VBC.

(Give the answer correct to 3 significant figures.)

The angle between the planes ABC and VBC is 60.5.

P. 25


B. Lines of Greatest SlopeB. Lines of Greatest Slope

Notes: There are infinitely many lines of greatest slope on a given inclined plane, such as l1, l2, and l3 (that are parallel to the line PQ) in the figure.

In the figure, the inclined plane ABCD intersects the horizontal plane ABEF at the line AB.

Three lines XY, PQ and ST are drawn on the inclined plane with PQ perpendicular to AB.

Let, and be the angles that XY, PQ and ST make with the horizontal plane respectively.

If we compare the three angles, we find that > and > .

In fact, PQ makes the largest angle with the horizontal plane and it is called the line of greatest slope of the inclined plane.

P. 26

Example 16.9T


cm 55 22 BP (Pyth. theorem)

cm 50

50

2tan NBP 0.2828

79.15NBP


The figure shows a right-angled triangular prism with ABCD,BCEF and AFED as rectangles. M and N are the mid-points of BC and AD respectively. If EC 5 cm, DE 2 cm and DA 5DE, find(a) the angle between the BN and plane BCEF,(b) the angle between line NC and plane BCEF,(c) the inclination of the line of greatest slope of plane ABCD.(Give the answers correct to 3 significant figures.)

Solution:BF 5 cm and AD FE 10 cm

(a) NBP is the required angle.

P Let P be the mid-point of EF.FP 5 cm and NP 2 cm

In BFP,

The required angle is 15.8.

P. 27

Example 16.9T


(cor. to 3 sig. fig.) 8.15

5

2tan DCE


The figure shows a right-angled triangular prism with ABCD,BCEF and AFED as rectangles. M and N are the mid-points of BC and AD respectively. If EC 5 cm, DE 2 cm and DA 5DE, find(a) the angle between the BN and plane BCEF,(b) the angle between line NC and plane BCEF,(c) the inclination of the line of greatest slope of plane ABCD.(Give the answers correct to 3 significant figures.)

Solution:

P

(b) NCP is the required angle.Since CPN BPN (SAS). NCP NBP

(c) Line of greatest slope: CD DCE is the required angle.

(cor. to 3 sig. fig.) 8.21DCE

P. 28

1166 ..33 Applications in Three-dimensionalApplications in Three-dimensional ProblemsProblems

In this section, we shall further study some applications of trigonometric formulas in three-dimensional figures, together with bearings and angles of elevation and depression.

P. 29

Example 16.10T

Solution:


In CDF,

CD

m 108sin

m 8sin

10

CD

m 71.853

In ACD, 222 CDADAC (Pyth. theorem)

m853.7165 22 AC 96.891 m

In ACE,

891.96

10sin ACE ACE 5.92

(cor. to 3 sig. fig.) The inclination of his path with the ground

EBCF is 5.92.

The figure shows the plane of a hillside. E is due west of F andC is due south of F. The inclination of the path CD is 8. DF 10 m and AD 65 m. Eric runs directly from C to A. (a) Find the inclination of his path with the ground EBCF. (b) Find the compass bearing of his path from C. (Give the answers correct to 3 significant figures.)

(a) Since EC is the projection of AC on the ground EBCF, ACE is the

required angle.

P. 30

Example 16.10T


In CDF,

CF

m 10tan8

m8tan

10

CF

71.1537 m

ECF 42.4122 42.4 (cor. to 3 sig. fig.)

Solution:

The figure shows the plane of a hillside. E is due west of F andC is due south of F. The inclination of the path CD is 8. DF 10 m and AD 65 m. Eric runs directly from C to A. (a) Find the inclination of his path with the ground EBCF. (b) Find the compass bearing of his path from C. (Give the answers correct to 3 significant figures.)

(b) The projection of AC on the ground EBCF is EC.

The compass bearing of Eric’s path from C is N42.4W.

1537.71

65tan ECF

In CEF, EF AD 65 m.

P. 31

Example 16.11T


PA

m09tan15

335.8846 m m 336

(cor. to 3 sig. fig.)

A lighthouse VA with height 90 m stands on the same plane astwo ships P and Q. The bearings of the lighthouse from P and Q are N50 E and N65 W respectively. The angle of elevation of V from P is 15 and the distance between P and Q is 800 m. (a) Find the distance between P and A.(b) Find the distance between Q and A. (Give the answers correct to 3 significant figures.)

Solution:(a) In VAP, (b) PAQ 50 65 115 (alt. s, // lines)

By cosine formula,

222 8846.335800 QA 115cos))(8846.335(2 QA05355.527181)(9019.2832 QAQA

m 598QA(cor. to 3 sig. fig.)

or 882 m (rejected)

15tan

m09PA

PAQQAPAQAPAQP cos))((2222

P. 32

Example 16.11T


8677.597

90tan VQA

VQA 8.5607 8.56 (cor. to 3 sig. fig.)

A lighthouse VA with height 90 m stands on the same plane astwo ships P and Q. The bearings of the lighthouse from P and Q are N50 E and N65 W respectively. The angle of elevation of V from P is 15 and the distance between P and Q is 800 m. (a) Find the distance between P and A.(b) Find the distance between Q and A.(c) Hence find the angle of elevation of V from Q. (Give the answers correct to 3 significant figures.)

Solution:(c) In VAQ,

The angle of elevation of V from Q is 8.56.

P. 33

16.1 Applications in Two-dimensional Problems

Chapter Chapter SummarySummary

1. Angles of elevation and depression

The angle between the line of sight of an object above us and the horizontal is the angle of elevation. The angle between the line of sight of an object below us and the horizontal is the angle of depression.

2. (a) Compass bearing

All directions are measured from the north (N) or the south (S). The bearing is expressed in the form N E, N W, S E or SW, where 0 90. (b) True bearing

All directions are measured from the north in a clockwise direction. The bearing is expressed in the form, where 0 360 and written in three digits.

P. 34

Chapter Chapter SummarySummary16.2 Basic Terminology in Three-dimensional Figures

1. Angle between Two Straight Lines

The angle between two intersecting straight lines is the acute angle formed by the two straight lines lying on the same plane.

2. Angle between a Straight Line and a Plane

The angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane.

3. Angle between Two Planes

The angle between two planes is the angle between two perpendiculars on the respective planes to the line of intersection of the two planes.

P. 35

Chapter Chapter SummarySummary16.2 Basic Terminology in Three-dimensional Figures

4. Distance between a Point and a Straight Line

The distance between a point and a straight line is the perpendicular distance from the point to the line.

5. Distance between a Point and a Plane

The distance between a point and a plane is the distance between the point and its projection on the plane.

6. Lines of Greatest Slope

If PQ AB, then PQ is called the line of greatest slope of the inclined plane ABCD.

P. 36

Chapter Chapter SummarySummary16.3 Applications in Three-dimensional Problems

In three-dimensional figures, we can also find (a) angles of elevation and depression, and (b) bearing.

1616 16.1Applications in Two-dimensional Problems 16.2Basic Terminology in Three-dimensional Figures...

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