Angles of elevation, angles of depression and bearings
In Year 12 Mathematical Methods (Queensland, QCAA), an angle of elevation is measured upwards from the horizontal and an angle of depression downwards from the horizontal; the two are equal by alternate angles. A bearing is a direction measured from north, clockwise — a three-figure true bearing such as \(035^\circ\text{T}\). These directions set up triangles you solve with right-angled trigonometry, the sine rule or the cosine rule.
The angle of elevation of an object is the angle between the horizontal and the line of sight when you look up at something higher than you. The angle of depression is the angle between the horizontal and the line of sight when you look down at something lower. Both are always measured from the horizontal, not from the vertical.
Because the horizontal line at the observer is parallel to the horizontal line at the object, the angle of depression from the observer equals the angle of elevation from the object back up to the observer — they are alternate angles between parallel lines. This equal-angle fact is the key to nearly every depression problem.
A bearing gives direction. A three-figure (true) bearing is measured from north, turning clockwise, and is written with three digits and a \(\text{T}\): due east is \(090^\circ\text{T}\), south \(180^\circ\text{T}\), west \(270^\circ\text{T}\). A compass bearing instead names the acute angle from north or south towards east or west, e.g. \(\text{S}\,50^\circ\text{E}\) (the same direction as \(130^\circ\text{T}\)). The back-bearing is the reverse direction, found by adding or subtracting \(180^\circ\).
Right-angled trigonometry (SOH–CAH–TOA) handles a single elevation or depression:
When the triangle is not right-angled, use the sine rule (a side and its opposite angle) or the cosine rule (two sides and the included angle):
Back-bearing (the reverse direction), kept between \(000^\circ\) and \(360^\circ\):
How to solve an elevation, depression or bearings problem
- Draw a clear diagram. Put in a north arrow at each relevant point; mark angles of elevation/depression from the horizontal and bearings clockwise from north.
- Transfer the directions to interior angles. Use alternate angles (depression = elevation), co-interior angles on parallel north lines, and back-bearings (\(\pm 180^\circ\)) to find the angles inside the triangle.
- Pick the right tool. A right angle → SOH–CAH–TOA. Two sides and the included angle → cosine rule. A complete opposite pair → sine rule.
- Substitute and solve. Keep full accuracy in the working; take the inverse trig function when finding an angle.
- Answer in context. Give a length to the stated precision and a bearing as a three-figure value \(\big(000^\circ\text{ to }360^\circ\big)\), or convert to compass form if asked.
Right-angled triangle; opposite \(=h\), adjacent \(=40\), so use \(\tan\).
| \(\tan 32^\circ\) | \(=\) | \(\dfrac{h}{40}\) |
| \(h\) | \(=\) | \(40\tan 32^\circ\) |
| \(=\) | \(25.0\ \text{m}\) |
The elevation from the boat equals the depression, \(12^\circ\) (alternate angles). Height \(=60\), distance \(=d\).
| \(\tan 12^\circ\) | \(=\) | \(\dfrac{60}{d}\) |
| \(d\) | \(=\) | \(\dfrac{60}{\tan 12^\circ}\) |
| \(=\) | \(282\ \text{m}\) |
Reverse the direction; since \(072^\circ<180^\circ\), add \(180^\circ\).
| \(\text{bearing of }A\text{ from }B\) | \(=\) | \(072^\circ+180^\circ\) |
| \(=\) | \(252^\circ\text{T}\) |
At \(B\), the reverse of \(AB\) is \(210^\circ\), so \(\angle ABC=210^\circ-150^\circ=60^\circ\) (included angle).
| \(AC^{2}\) | \(=\) | \(5^{2}+8^{2}-2(5)(8)\cos 60^\circ\) |
| \(=\) | \(89-40=49\) | |
| \(AC\) | \(=\) | \(7\ \text{km}\) |
Common pitfalls
Frequently asked questions
What is an angle of elevation?
The angle measured upwards from the horizontal to your line of sight when you look up at an object higher than you.
What is an angle of depression?
The angle measured downwards from the horizontal to an object below you. It equals the angle of elevation from that object back up to you, because the two horizontals are parallel (alternate angles).
How is a three-figure bearing measured?
From north, turning clockwise, written with three digits and a T: \(035^\circ\text{T}\), \(210^\circ\text{T}\). East is \(090^\circ\), south \(180^\circ\), west \(270^\circ\).
How do you find a back-bearing?
Add \(180^\circ\) if the bearing is under \(180^\circ\), otherwise subtract \(180^\circ\), so the answer stays between \(000^\circ\) and \(360^\circ\). E.g. the reverse of \(072^\circ\) is \(252^\circ\).
When do you use the sine or cosine rule?
After marking the interior angles from the bearings: right-angled trig if there is a right angle, the cosine rule for two sides and the included angle, the sine rule for a side and its opposite angle plus one more fact.
What is the difference between a true and a compass bearing?
A true bearing is one clockwise angle from north (\(130^\circ\text{T}\)); a compass bearing names the acute angle from north or south towards east or west (\(\text{S}\,50^\circ\text{E}\) is the same direction).