Transcript 3D Trigonometry

Starter
Can you find the value of x in this diagram to 2dp? Be
careful with rounding answers!
1) Start by finding the opposite
side in the left triangle
NOT TO SCALE
Hyp
𝑂𝑝𝑝 = 𝑇𝑎𝑛𝜃 × 𝐴𝑑𝑗
13cm
x
49.82°
42°
8cm
Adj
𝑂𝑝𝑝 = 𝑇𝑎𝑛42 × 8
5.93
4cm
5.93
24°
𝑆𝑖𝑛𝜃 =
𝑂𝑝𝑝
𝐴𝑑𝑗
𝑆𝑖𝑛𝜃 =
9.93
13
𝑂𝑝𝑝 = 7.20 (2𝑑𝑝)
Opp
7.20
3) Then you can find the angle x by
using the opposite and hypotenuse in
the upper triangle
Opp
Hyp
12cm
Adj
2) Then subtract this from the
hypotenuse of the lower triangle
𝐻𝑦𝑝 =
𝐴𝑑𝑗
𝐶𝑜𝑠𝜃
𝐻𝑦𝑝 =
12
𝐶𝑜𝑠24
𝐻𝑦𝑝 = 13.14 (2𝑑𝑝)
𝑆𝑞𝑢𝑎𝑟𝑒 𝑠𝑖𝑑𝑒 = 13.14 − 7.20 = 5.93
𝑆𝑖𝑛𝜃 = 0.764 …
𝜃 = 49.82°
3D Trigonometry
• We have looked at using trigonometry in lots of
situations
• Today we will be focusing on using it in 3D
problems
• The key to answering questions in 3D is thinking
about separate parts of them in 2D
• You will find that making quick sketches as you
work will help you visualise what to do!
3D Trigonometry
ABCDEFGH is a cuboid, as shown.
a) Calculate the length of BD 8.06cm (√65)
b) Calculate the angle between BH and the base of the cuboid
 You can find the length of BD by just using the base
H
G
4cm
E
6cm
A
F
D
7cm
C
4cm
7cm
B
𝑎2 + 𝑏 2 = 𝑐 2
72 + 42 = 𝑐 2
65 = 𝑐 2
Sub in values
Add up
Square root
8.06 = 𝑐
(Remember that as an exact value, c = √65!)
3D Trigonometry
ABCDEFGH is a cuboid, as shown.
a) Calculate the length of BD 8.06cm (√65)
b) Calculate the angle between BH and the base of the cuboid
 To find the angle between BH and the base, draw on BH,
and look to make a right angled triangle (inside the shape)
H
G
Opp
6
E
6cm
F
D
√65
A
θ
C
7cm
4cm
θ
B
√65
𝑂𝑝𝑝
𝑇𝑎𝑛𝜃 =
𝐴𝑑𝑗
𝑇𝑎𝑛𝜃 =
6
65
𝑇𝑎𝑛𝜃 = 0.744 …
𝜃 = 36.7°
Adj
Sub in values
Calculate
Use inverse Tan
Plenary
The great pyramid of Giza is a square based pyramid of side length 230m,
and is 139m high. If you were to stand in one corner and walk up the
pyramid to the top, what angle would you be walking up at and how far
would you have to walk?
 We need the length of the diagonal first
𝑎2 + 𝑏 2 = 𝑐 2
230m
2302 + 2302 = 𝑐 2
105800 = 𝑐 2
230m
325.27m
230m
230m
325.27 = 𝑐
Sub in values
Add up
Square root
Plenary
The great pyramid of Giza is a square based pyramid of side length 230m,
and is 139m high. If you were to stand in one corner and walk up the
pyramid to the top, what angle would you be walking up at and how far
would you have to walk?
 Now we can draw the height on, and halve the length
we just found…
139m
𝑇𝑎𝑛𝜃 =
Opp
139m
325.27m
162.63m
230m
230m
40.5°
θ
𝑂𝑝𝑝
𝐴𝑑𝑗
139
𝑇𝑎𝑛𝜃 =
162.63
𝑇𝑎𝑛𝜃 = 0.854 …
𝜃 = 40.5°
Sub in values
Calculate
Use inverse Tan
162.63m
Adj
So you would be walking
up at an angle of 40.5°
Plenary
The great pyramid of Giza is a square based pyramid of side length 230m,
and is 139m high. If you were to stand in one corner and walk up the
pyramid to the top, what angle would you be walking up at and how far
would you have to walk?
 Now we can draw the height on, and halve the length
we just found…
So you would be walking up at an
angle of 40.5°
139m
139m
230m
162.63m
230m
40.5°
162.63m
𝑎2 + 𝑏 2 = 𝑐 2
162.632 + 1392 = 𝑐 2
45771 = 𝑐 2
Sub in values
Add up
Square root
213.9 = 𝑐
So you would have to walk a distance
of 213.9m to the top!
Summary
• We have looked at using Trigonometry
in 3D shapes
• We have seen how to model this using
2D diagrams
• We have seen that diagrams help a lot!!
Starter (printout)
Can you find the value of x in this diagram to 2dp? Be
careful with rounding answers!
NOT TO SCALE
13cm
x
42°
4cm
8cm
24°
12cm
Plenary (printout)
The great pyramid of Giza is a square based pyramid of side length 230m,
and is 139m high. If you were to stand in one corner and walk up the
pyramid to the top, what angle would you be walking up at and how far
would you have to walk?