HKDSE MATHEMATICS Ronald Hui Tak Sun Secondary School

HOMEWORK  SHW6-01, 6-A1  Deadline: 11 Jan 2016  No more delay please 22 October 2015 Ronald HUI

SUMMARY OF 2D TRIGONOMETRY Ronald Hui Tak Sun Secondary School

Note: The above formula is also valid for right-angled triangles. When C = 90 , 2 1 ab sin C = = bc sin A 2 1 = ac sin B 2 1 the included angle of a and b. Similarly, for any △ ABC, we have: Area of △ ABC area of △ ABC the included angle of b and c. the included angle of a and c. b c A B a C b a C b c A c a B

Area of △ ABC where s s is called the semi-perimeter. Heron’s formula s(s  a)(s  b)(s  c), A C a b c B

A a B b C c Note: By the sine formula, we have: 1. 2.sin A : sin B : sin C = a : b : c The sine formula In △ ABC, a sin A = b sin B = c sin C It can also be written as = b sin B = c sin C sin A a

The cosine formula The above results are known as the cosine formula. In △ ABC, In fact, for any △ ABC, we have c 2 = a 2 + b 2  2ab cos C Similarly, we can prove that b 2 = a 2 + c 2  2ac cos B and a 2 = b 2 + c 2  2bc cos A. a 2 = b 2 + c 2  2bc cos A, b 2 = a 2 + c 2  2ac cos B, c 2 = a 2 + b 2  2ab cos C. A C a b c B

In △ ABC, A C a b c B The cosine formula can also be written as follows: The cosine formula is also known as the cosine law or the cosine rule.

horizontal line A B C D Angle of elevation of B from A Angle of depression of A from B = angle of elevation of B from A In the figure, CB and AD are horizontal lines. We can see that: angle of depression of A from B alt.  s, AD // CB

To describe the direction of a point relative to another point, true bearing or compass bearing may be used. True Bearing and Compass Bearing True Bearing N O P Directions are measured from the north in a clockwise direction. It is expressed as x , where (i)0  x < 360, (ii)the integral part of x must consist of 3 digits. N O P Q 58  40  Refer to the figure on the right, the true bearing of Q from O is the true bearing of P from O is 180  + 40  = 220  058 , 220 .

S N Compass Bearing O P It is expressed as Nx  E, 40  N Directions are measured from the north W E N40  E 60  25  Q R S N60  W S70  W 70  S25  E S or the south. NxW,NxW, SxESxEor Sx  W, where 0 < x < 90. True Bearing and Compass Bearing

Book 5A Chapter 6 Basic Terminologies in 3-dimensional Problems

Angle between Two Intersecting Straight Lines The figure shows two non-parallel straight lines AB and CD. They intersect each other at a point E.   E P R Q S 90 . If the two lines are perpendicular to each other, then the angle between the two lines is A B D C The acute angle  formed is called the angle between two straight lines.

Angle between Two Intersecting Straight Lines The figure shows two non-parallel straight lines AB and CD. They intersect each other at a point E.   E A B D C How about ∠ AED and ∠ CEB? Are they also called the angle between two straight lines? The acute angle  formed is called the angle between two straight lines.

Angle between Two Intersecting Straight Lines The figure shows two non-parallel straight lines AB and CD. They intersect each other at a point E.   E A B D C No, these obtuse angles are NOT considered as the angle between the two straight lines. The acute angle  formed is called the angle between two straight lines.

∵ AC and AG intersect at. A A B C D E H G F ∴ The angle between the lines AC and AG is. ∠ GAC Can you identify the angle between the lines AC and AG?

Follow-up Question The figure shows a cube ABCDEFGH. Identify the angles between (a) the lines BG and GH, (b) the lines AD and BD, (c) the lines AC and CF. A B C D E F G H  BGH. (a) ∵ BG and GH intersect at ∴ The angle between the lines BG and GH is G.G.

Follow-up Question The figure shows a cube ABCDEFGH. Identify the angles between A B C D E F G H (b) Join BD. ∴ The angle between the lines AD and BD is ∵ AD and BD intersect at D.D.  ADB. (a) the lines BG and GH, (b) the lines AD and BD, (c) the lines AC and CF.

Follow-up Question The figure shows a cube ABCDEFGH. Identify the angles between A B C D E F G H (c) Join AC and CF. ∴ The angle between the lines AC and CF is ∵ AC and CF intersect at C.  ACF. (a) the lines BG and GH, (b) the lines AD and BD, (c) the lines AC and CF.

PQ is perpendicular to any straight line on  passing through Q (e.g. L 1 and L 2 ). In the figure, P is a point lying outside the plane .  P Q is a point on the plane  such that Q L2L2 L1L1 (ii)Q is the projection of P on the plane . Then, (i) PQ  plane  Angle Between a Straight Line and a Plane

In the figure, a straight line AB intersects the plane  at the point A. Angle Between a Straight Line and a Plane  B A C C is the projection of B on the plane . (ii)  BAC is the angle between the line AB and the plane . (i)AC is the projection of AB on the plane . Then,

Can you identify the angle between the line FD and the plane ABCD ? A B C D E H G F ∵ is the projection of FD on the plane ABCD. ∴ The angle between the line FD and the plane ABCD is. BD ∠ FDB

Follow-up question The figure shows a right triangular prism with ∠ ABC = ∠ DEF = 90 . Identify the angles between (a)the line AF and the plane ADEB. (a)Join AF. ∴ The angle between the line AF and the plane ADEB is (b)the line AF and the plane DEF. A B C D E F ∵ AE is the projection of AF on the plane ADEB. ∠ FAE.

Follow-up question The figure shows a right triangular prism with ∠ ABC = ∠ DEF = 90 . Identify the angles between (a)the line AF and the plane ADEB. (b) ∵ DF is the projection of AF on the plane DEF. ∴ The angle between the line AF and the plane DEF is (b)the line AF and the plane DEF. A B C D E F ∠ AFD.

After identifying angles in three dimensions, we can find their sizes by applying trigonometric knowledge. Let’s see how to solve the following problems. A B C D E H G F 10 cm 6 cm Angle between the lines CH and CD = ? Angle between the line AF and the plane ABCD = ? A B F E C D 12 cm 4 cm 5 cm

A B C D E H G F 10 cm 6 cm Step 1:Identify the required angle. ∴ The angle between the lines CD and CH is ∵ CD and CH intersect at  DCH. The figure shows a rectangular block ABCDHEFG. AB = 10 cm and AE = 6 cm. Find the angle between the lines CD and CH, correct to 1 decimal place. C.C.

Step 2:Analyse the information needed to find the required angle. Do you know the lengths of any side of △ CDH? Yes, I know the lengths of CD and DH. A B C D E H G F 10 cm 6 cm The figure shows a rectangular block ABCDHEFG. AB = 10 cm and AE = 6 cm. Find the angle between the lines CD and CH, correct to 1 decimal place.

CD = AB = 10 cm and DH = AE = 6 cm Then you can find ∠ DCH by using trigonometric ratio. Step 2:Analyse the information needed to find the required angle. A B C D E H G F 10 cm 6 cm The figure shows a rectangular block ABCDHEFG. AB = 10 cm and AE = 6 cm. Find the angle between the lines CD and CH, correct to 1 decimal place.

The angle between the lines CD and CH is ∠ DCH. Solution DC = AB = 10 cm DH = AE = 6 cm Consider △ CDH. 10 cm 6 cm  tan  DCH DC DH   DCH  31.0  (cor. to 1 d.p.) ∴ The angle between the lines CD and CH is 31.0 . A B C D E H G F 10 cm 6 cm The figure shows a rectangular block ABCDHEFG. AB = 10 cm and AE = 6 cm. Find the angle between the lines CD and CH, correct to 1 decimal place.

In some cases, Pythagoras theorem is also useful in finding the angles.

The figure shows a right triangular prism ABCDEF. AB = 12 cm, BC = 5 cm, CF = 4 cm and ∠ BCF = ∠ ADE = 90 . Find the angle between the line AF and the plane ABCD, correct to 3 significant figures. A B F E C D 12 cm 4 cm 5 cm ∴ The angle between the line AF and the plane ABCD is ∵ AC is the projection of AF on the plane ABCD. ∠ FAC. Step 1:Identify the required angle.

Consider △ ABC, we can use Pythagoras theorem to find the length of AC. Step 2:Analyse the information needed to find the required angle. In △ ACF, CF = 4 cm AC = ? AF = ? Then we have enough information to find ∠ FAC. A B F E C D 12 cm 4 cm 5 cm The figure shows a right triangular prism ABCDEF. AB = 12 cm, BC = 5 cm, CF = 4 cm and ∠ BCF = ∠ ADE = 90 . Find the angle between the line AF and the plane ABCD, correct to 3 significant figures.

The required angle is  FAC. Solution Consider △ ABC. AC AB 2 + BC 2   cm ◄ Pyth. theorem  13 cm Consider △ ACF. 13 cm 4 cm  tan  FAC AC CF   FAC  17.1  (cor. to 3 sig. fig.) ∴ The angle between the line AF and the plane ABCD is 17.1 . A B F E C D 12 cm 4 cm 5 cm The figure shows a right triangular prism ABCDEF. AB = 12 cm, BC = 5 cm, CF = 4 cm and ∠ BCF = ∠ ADE = 90 . Find the angle between the line AF and the plane ABCD, correct to 3 significant figures.

Follow-up question A B C D E H G F 8 cm 7 cm 6 cm The figure shows a rectangular block ABCDHEFG. AB = 8 cm, BC = 6 cm and AE = 7 cm. Find the angle between the line CE and the plane ABCD, correct to 3 significant figures. ∴ The angle between the line CE and the plane ABCD is ∠ ECA. ∵ CA is the projection of CE on the plane ABCD. Consider △ ABC. AC AB 2 + BC 2   cm ◄ Pyth. theorem  10 cm

Follow-up question Consider △ ACE. 10 cm 7 cm  tan  ECA AC AE   ECA  35.0  (cor. to 3 sig. fig.) ∴ The angle between the line CE and the plane ABCD is 35.0 . A B C D E H G F 8 cm 7 cm 6 cm The figure shows a rectangular block ABCDHEFG. AB = 8 cm, BC = 6 cm and AE = 7 cm. Find the angle between the line CE and the plane ABCD, correct to 3 significant figures.

Angle Between Two Intersecting Planes intersects plane  1 It is given that plane  2 at a straight line AB. 11 AB is the line of intersection of the two planes. 22 B A

Angle Between Two Intersecting Planes If PX is a line on plane  1 such that PX  AB and 22 11 X Y B A P PY is a line on plane  2 such that PY  AB, then P is a point on AB.  The acute angle  between PX and PY is the angle between the planes  1 and  2.

Can you identify the angle between the planes ADHE and AFGD? Do you notice that DH ⊥ AD and DG ⊥ AD? A B C D E H G F is the line of intersection of the planes ADHE and AFGD. AD ∴ The angle between the planes ADHE and AFGD is ∠ EAF. FA ⊥ AD ∵ AE ⊥ AD and Therefore, the required angle can also be ∠ HDG. (or ∠ HDG)

If AE = 14 cm and AF = 20 cm, can you find the angle between the planes ADHE and AFGD, correct to 3 significant figures? A B C D E H G F Consider △ AEF. 20 cm 14 cm  cos  EAF AF AE   EAF  45.6  (cor. to 3 sig. fig.) ∴ The angle between the planes ADHE and AFGD is 45.6 . 14 cm 20 cm The required angle is ∠ EAF.

(a) ∵ BF ⊥ EF Follow-up question ∴ The angle between the planes BEF and CDEF is A B C D FE and CF ⊥ EF  BFC. EF is the line of intersection of the planes BEF and CDEF. The figure shows a triangular prism ABCDEF, where ABCD, CFED and ABFE are rectangles. It is given that AE = 12 cm, BC = 10 cm and ∠ BCF = 90 . (a) Identify the angle between the planes BEF and CDEF. (b) Find the angle mentioned in (a), correct to 3 significant figures. 12 cm 10 cm

(b) Follow-up question ∴ The angle mentioned in (a) (i.e.  BFC) is 56.4 . A B C D FE The figure shows a triangular prism ABCDEF, where ABCD, CFED and ABFE are rectangles. It is given that AE = 12 cm, BC = 10 cm and ∠ BCF = 90 . (a) Identify the angle between the planes BEF and CDEF. (b) Find the angle mentioned in (a), correct to 3 significant figures. 12 cm 10 cm Consider △ BCF. 12 cm 10 cm  sin  BFC BF BC   BFC  56.4  (cor. to 3 sig. fig.) BF = AE = 12 cm

I will choose the path that is less steep. Let us compare the slopes of the paths AX and BX. Line of Greatest Slope The figure shows an inclined road with two paths AX and BX. B A X Y Which path do you choose to reach X?

Line of Greatest Slope Consider  BXY. Consider  AXY. ∵ AY  BY i.e.Slope of BX < slope of AX B A Y Slope of BX = BY XY Slope of AX = AY XY ∴ AYBY 11  ∴ AY XY BY XY  Here, we can see that different straight lines on an inclined road have different slopes. Now, let us study how to find the line with the greatest slope on an inclined plane. So, I will choose the path BX to reach X. X Slope of line segment vertical distance horizontal distance = X

Line of Greatest Slope The figure shows an inclined plane ABEF. It intersects the horizontal plane ABCD at AB. If PX  AB, the slope of PX is the greatest. X P PX is the line of greatest slope of the inclined plane ABEF.

A B E F C D 15 cm 4 cm The figure shows a hillside ABCD sloping to the horizontal ground ABEF. ABCD is a rectangle. It is given that CE = 4 cm and BE = 15 cm and ∠ BEC = 90 . Find the inclination of the line of greatest slope of the hillside ABCD. (Give your answer correct to 3 significant figures.)

∴ BC is the line of greatest slope of the hillside ABCD. A B E F C D 15 cm 4 cm The figure shows a hillside ABCD sloping to the horizontal ground ABEF. ABCD is a rectangle. It is given that CE = 4 cm and BE = 15 cm and ∠ BEC = 90 . ∵ AB is the line of intersection of the inclined plane ABEF and the horizontal plane ABCD, and BC ⊥ AB.

15 cm 4 cm   CBE  14.9  (cor. to 3 sig. fig.) tan  CBE BE CE  Consider △ BCE. ∴ The inclination of the line of greatest slope of the hillside ABCD is 14.9 . A B E F C D 15 cm 4 cm The figure shows a hillside ABCD sloping to the horizontal ground ABEF. ABCD is a rectangle. It is given that CE = 4 cm and BE = 15 cm and ∠ BEC = 90 .

Follow-up question The figure shows a rectangular board ABCD sloping to a horizontal plane. E and F are the projections of the points C and D on the horizontal plane respectively. It is given that CE = 5 cm, AB = 18 cm and  BAC = 38 . (a)Find the length of the line of greatest slope of the plane ABCD. (b)Find the inclination of the line of greatest slope of the plane ABCD. A B E F CD 18 cm 38  5 cm (Give your answers correct to 3 significant figures.)

A B E F C D 18 cm 38  5 cm tan 38  18 cm BC  BC  18 tan 38  cm  14.1 cm (cor. to 3 sig. fig.) tan  BAC AB BC  ∴ The length of the line of greatest slope is 14.1 cm. (a) BC is the line of greatest slope of the plane ABCD. Consider △ ABC.

18 tan 38  cm 5 cm  sin ∠ CBE BC CE  ∠ CBE 20.8  (cor. to 3 sig. fig.)  (b) ∠ CBE is the inclination of the line of greatest slope of the plane ABCD. Consider △ BCE. ∴ The inclination of the line of greatest slope of the plane ABCD is 20.8 . BC  18 tan 38  cm A B E F C D 18 cm 38  5 cm

Consider a straight line L and a point P not lying on L. Distance between a Point and a Line If Q is a point on L such that PQ is perpendicular to the line L, then, L P Q Note:PQ is also the shortest distance between the point P and the line L. PQ is the distance between the point P and the line L.

The distance between the point F and the line AB = A B C D E F G H 9 cm 8 cm 7 cm Example: The distance between the point B and the line CD = The distance between the point A and the line BG = The figure shows a cuboid ABCDEFGH. It is given that AB = 9 cm, BC = 8 cm and AF = 7 cm. 7 cm 8 cm 9 cm

Consider a point R not lying on the plane , and its projection, say point S on the plane . Distance between a Point and a Plane  R S Note:RS is also the shortest distance between point R and the plane . Then, RS is perpendicular to the plane , and RS is the distance between the point R and the plane .

The figure shows a rectangular board ABCD inclining at 25  to the horizontal plane. E and F are the projections of the points C and D on the horizontal plane respectively. It is given that BC = 15 cm. A B E F C D 15 cm 25  Find the distance between the point B and the plane CDFE. (Give your answer correct to 3 significant figures.)

The figure shows a rectangular board ABCD inclining at 25  to the horizontal plane. E and F are the projections of the points C and D on the horizontal plane respectively. It is given that BC = 15 cm. Consider △ BCE. cos 25  15 cm BE  cos  CBE BC BE  ∴ BE is the distance between the point B and the plane CDFE. ∵ E is the projection of B on the plane CDFE. BE  13.6 cm (cor. to 3 sig. fig.) ∴ The distance between the point B and the plane CDFE is 13.6 cm. A B E F C D 15 cm 25 

Follow-up question The figure shows a rectangular block ABCDEFGH. It is given that EH = 30 cm and ∠ HFG = 20 . Find the distances between (a)the point G and the line EH, (b)the point G and the plane ACHF. A B C D E F G H 30 cm 20  (a) Consider △ FGH. tan 20  30 cm GH   FG = EH = 30 cm tan  HFG FG GH  ∴ GH is the distance between the point G and the line EH. ∵ GH ⊥ EH (Give your answers correct to 3 significant figures.)

 10.9 cm (cor. to 3 sig. fig.) ∴ The distance between the point G and the line EH is 10.9 cm. GH  30 tan 20  cm Consider △ FQG. (b) Let Q be the projection of point G on the plane ACHF. sin 20  30 cm QG  sin  GFQ FG QG   10.3 cm (cor. to 3 sig. fig.) ∴ The distance between the point G and the plane ACHF is 10.3 cm. A B C D E F G H 30 cm 20  Then GQ is the distance between the point G and the plane ACHF. Q 30 sin 20  cm QG 