Trigonometry

3D Trigonometry

r

s

h

p

qβ

α

p, q and r are points on level ground, [sr] is a vertical flagpole of height h. The angles of elevation of the top of the flagpolefrom p and q are α and β, respectively.

(i) If | α | = 60º and | β | = 30º, express | pr | and | qr | in terms of h.

30º

60º

s

h

rp ADJ

OPP

tan 60h

pr

3h

pr

3

hpr

s

h

r

p

q30º

60º

60º

tan30h

qr

1

3

h

qr 3qr h

s

h

r

p

qOPP

ADJ

30º

60º

r

s

h

p

q3h

(ii) Find | pq | in terms of h, if tan qrp = 8.

3

h

a2 = b2 + c2 – 2bccosAA

8

1

2 2 21 81 89

( )x

x

Pythagoras’ Theorem

3

1cos

3A

30º

60º

r

p

q

3

h

a2 = b2 + c2 – 2bccosA

2

22 13 2 3

33 3

h hqp h h

2

2 223

3 3

hh h

2 2 29 2

3

h h h

28

3

h

28

3

hqp

8

3h

1cos

3A

slanted edge

The great pyramid at Giza in Egypt has a square base and four triangular faces.

The base of the pyramid is of side 230 metres and the pyramid is 146 metres high.

The top of the pyramid is directly above the centre of the base.

(i) Calculate the length of one of the slanted edges, correct to the

nearest metre.

230 m

230 m

Pythagoras’ theoremx

2 2 2230 230x

105800

2

325·269..

x

2

x 162·6

162·6

162·6

146

2006 Paper 2 Q5 (b)

slanted edge

The great pyramid at Giza in Egypt has a square base and four triangular faces.

The base of the pyramid is of side 230 metres and the pyramid is 146 metres high.

The top of the pyramid is directly above the centre of the base.

(i) Calculate the length of one of the slanted edges, correct to the

nearest metre.

146 m

162·6 m

Pythagoras’ theoreml

47754·76

2

l 218·528..

l

219 m162·6

146

2 2 2146 162·6l

2006 Paper 2 Q5 (b)

(ii) Calculate, correct to two significant numbers, the total area of the four triangular faces of the pyramid (assuming they are smooth flat surfaces)

slanted edge

219 m

230 m 347362 186·375..

h 186·4 m

h

115 m

Pythagoras’ theorem2 2 2219 115h

2 2 2219 115h

Area of triangle base × height

12

(230)(186·4)

12

21436 m2

2006 Paper 2 Q5 (b)

slanted edge

(ii) Calculate, correct to two significant numbers, the total area of the four triangular faces of the pyramid (assuming they are smooth flat surfaces)

219 m

347362 186·375..

h 186·4 m

h

115 m

Pythagoras’ theorem2 2 2219 115h

2 2 2219 115h

Total area 21436 4

85744 m2 86000 m2

2006 Paper 2 Q5 (b)

θ

2θ

3x

x

q

pqrs is a vertical wall of height h on level ground. p is a point on the ground in front of the wall. The angles of elevation of r from p is θ and the angle of elevation of s from p is 2θ.

| pq | = 3| pt |.

Find θ.

p

s

r

h

tθ3x qp

r

h

tan3

h

x 3 tanh x

2005 Paper 2 Q5 (c)

θ

2θ

3x

x

q

pqrs is a vertical wall of height h on level ground. p is a point on the ground in front of the wall. The angles of elevation of r from p is θ and the angle of elevation of s from p is 2θ.

| pq | = 3| pt |.

Find θ.

p

s

r

h

t2θ

x tp

s

h

tan 2h

x tan 2h x

2005 Paper 2 Q5 (c)

2

2tan

1 tan

tan θx θ

2θ

3x

x

qp

s

r

h

t2

2tantan 2

1 tan

tan 2θx3

Let t = tan θ

2

23

1

tt

t

23 1 2t t t

33 0t t 21 3 0t t

0t

2 1

3t

1tan

3t

33 3 2t t t

21 3 0t

6

2005 Paper 2 Q5 (c)

a

c

d

(i) Find | bac | to the nearest degree.

4.D Question 4

b 5

m 5

2

4

A

2 2 24 5 5 2(5)(5)cos A

2 2 2 2 cosa b c bc A

16 25 25 50cos A

16 50 50cos A 50cos 50 16A

34cos

50A 47·156....A

abc is an isosceles triangle on a horizontal plane, such that |ab| |ac| 5 and |bc| 4.m is the midpoint of [bc].

47A

abc is an isosceles triangle on a horizontal plane, such that |ab| |ac| 5 and |bc| 4.m is the midpoint of [bc]. a

c

d

4.D Question 4

b 5

m 5

2

(ii) A vertical pole [ad] is erectedat a such that |ad | 2, find|amd | to the nearest degree.

2

2 2 22 5am 225 4am

23·578..amd 24

221am

21

amd

amd 2tan

21