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.

30º60º

s

h

rp ADJ

OPP

tan 60 hpr

3 hpr

3

hpr

s

h

r

p

q30º

60º

60º

tan30 hqr

13

hqr

3qr h

s

h

r

p

qOPP

ADJ30º

60º

r

s

h

p

q3h

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

3h

a2 = b2 + c2 – 2bccosAA

8

1

2 2 21 81 89

( )x

x

Pythagoras’ Theorem

3

1cos3

A 30º

60º

r

p

q

3h

a2 = b2 + c2 – 2bccosA

2

22 13 2 333 3

h hqp h h 2

2 2233 3hh h

2 2 29 23

h h h

283h

283hqp

83

h

1cos3

A

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

2x 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)

θ2θ

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

htθ

3x qp

r

h

tan3hx

3 tanh x

2005 Paper 2 Q5 (c)

θ2θ

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

ht

2θx tp

s

h

tan 2 hx

tan 2h x

2005 Paper 2 Q5 (c)

2

2tan1 tan

tan θx θ2θ

3x

x

qp

s

r

ht

2

2tantan 21 tan

tan 2θx3

Let t = tan θ

2

231

ttt

23 1 2t t t

33 0t t 21 3 0t t

0t

2 13

t 1tan3

t

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

4A

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

34cos50

A 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 2 25 4am

23·578..amd 24

2 21am

21

amd

amd 2tan21