2D and 3D Trigonometry Problems | IEB Grade 12 Maths

Problems in two and three dimensions ask you to find heights, distances and angles by working through several triangles in turn. The IEB often asks you to prove a general formula first and then use it with numbers, so algebra with trig ratios matters here too.

Angles of elevation and depression

The angle of elevation is measured up from the horizontal to the line of sight.
The angle of depression is measured down from the horizontal to the line of sight.

The angle of depression from $A$ to $B$ equals the angle of elevation from $B$ to $A$ (alternate angles, since the horizontals are parallel).

Example. From the top of a 40 m building, the angle of depression to a car is 28°. The distance from the car to the foot of the building is

$d = \frac{40}{t a n 28°} \approx 75, 23 m$

Draw a clear diagram, mark the right angles, and label the horizontal before placing any angle.

Multi-triangle problems in 3D

In 3D problems, a vertical object (a tower or pole) stands on a horizontal plane, and you work in two triangles:

a horizontal triangle on the ground, solved with the sine, cosine or area rule, and
a vertical right-angled triangle, solved with SOH CAH TOA.

Strategy:

Identify the side the two triangles share. That is your bridge.
Find the shared side in the triangle where you have enough information.
Use it in the second triangle.

Example. A flagpole $T P$ stands at $P$ . From points $A$ and $B$ on the ground, 50 m apart, $\hat{P A B} = 40°$ and $\hat{P B A} = 65°$ . The angle of elevation of $T$ from $A$ is 20°.

In $△ A B P$ : $\hat{A P B} = 75°$ , so $A P = \frac{50 sin 65°}{sin 75°} \approx 46, 91$ m.

In $△ A T P$ : $T P = A P tan 20° \approx 17, 08$ m.

Measurement can be tested here as well, such as the volume of a cone or the surface area of a prism formed in the diagram.

Key terms

Angle of elevation: The angle above the horizontal from an observer to an object.
Angle of depression: The angle below the horizontal from an observer to an object.
Horizontal plane: The flat ground in a 3D problem, perpendicular to any vertical object.

Common mistakes

Measuring the angle of depression from the vertical instead of the horizontal.
Using SOH CAH TOA in a triangle that is not right-angled.
Treating a 3D diagram as if all triangles lie in the same plane.
Rounding the shared side before using it in the next triangle.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

From the top of a building 40 m high, the angle of depression to a car on the road is 28°. Calculate the distance from the car to the foot of the building.

Worked solution

Hidden - try the question first

Solution

The angle of elevation from the car to the top is 28° (alternate angles)
$tan 28° = \frac{40}{d}$
$d = \frac{40}{tan 28°}$
$d \approx 75, 23$ m

Question 2

6 marks total

routine

Two observers, $A$ and $B$ , stand 100 m apart on level ground in a straight line with the foot $F$ of a tower $T F$ , with $B$ closer to the tower. The angles of elevation of $T$ from $A$ and $B$ are 30° and 45°.

Determine the height of the tower, in simplest surd form and to two decimal places.

Worked solution

Hidden - try the question first

Solution

$B F = \frac{h}{tan 45°} = h$
$A F = \frac{h}{tan 30°} = h 3$
$A F - B F = 100$ : $h 3 - h = 100$
$h = \frac{100}{3 - 1}$
$= 50 (3 + 1)$
$\approx 136, 60$ m

Question 3

7 marks total

complex

A vertical flagpole $T P$ stands at point $P$ on level ground. Points $A$ and $B$ are on the ground with $A B = 50$ m, $\hat{P A B} = 40°$ and $\hat{P B A} = 65°$ . The angle of elevation of $T$ from $A$ is 20°.

(a) Calculate the length of $A P$ . (4)

(b) Calculate the height of the flagpole. (3)

Worked solution

Hidden - try the question first

Solution

(a) $\hat{A P B} = 180° - 40° - 65° = 75°$
$\frac{A P}{sin 65°} = \frac{50}{sin 75°}$
$A P = \frac{50 sin 65°}{sin 75°}$
$A P \approx 46, 91$ m
(b) $tan 20° = \frac{T P}{A P}$
$T P = 46, 91 tan 20°$
$T P \approx 17, 08$ m

Question 4

9 marks total

problem-solving

A vertical tower $T P$ of height $h$ stands at $P$ . Points $A$ and $B$ lie on level ground with $P$ , and $A B = d$ . $\hat{P A B} = α$ , $\hat{P B A} = β$ , and the angle of elevation of $T$ from $A$ is $θ$ .

(a) Prove that $h = \frac{d sin β tan θ}{sin ( α + β )}$ . (5)

(b) Calculate $h$ if $d = 80$ m, $α = 52°$ , $β = 61°$ and $θ = 24°$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $\hat{A P B} = 180° - 40° - 65° = 75°$
$\frac{A P}{sin β} = \frac{d}{sin ( 180° - ( α + β ))}$
$sin (180° - (α + β)) = sin (α + β)$ , so $A P = \frac{d sin β}{sin ( α + β )}$
In $△ T A P$ : $h = A P tan θ$
$h = \frac{d sin β tan θ}{sin ( α + β )}$
(b) $A$
$h \approx 33, 84$ m
(c) $B P = \frac{80 sin 52°}{sin 113°}$
$\approx 68, 49$ m

FAQs

How do I approach a 3D trig problem?

Find the side shared by the horizontal triangle and the vertical triangle. Solve the triangle where you have enough information first, then use the shared side in the other.

Why does the IEB ask me to prove a formula first?

It tests whether you can work with trig ratios algebraically. If you cannot do the proof, you can still use the given formula in the next part.

Can measurement questions appear in trigonometry?

Yes. The IEB can include surface area or volume in a 2D or 3D trig question, using shapes formed in the diagram.

Problems in two and three dimensions

Angles of elevation and depression

Multi-triangle problems in 3D

Key terms

Common mistakes

Practice questions

FAQs

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