Angle of Inclination and Angle of Depression Explained (With Examples)
- Tyler Buffone
- Sep 21
- 5 min read
Updated: Oct 16
Angle of Inclination
It's how much a line is tilted above the horizontal.
Picture a flat ground line. If a ramp or road rises up from that flat line, the angle it makes with the flat line is the angle of inclination.
In the following drawing, the angle of inclination is indicated by the symbol theta (θ).
Angle of Depression
This is an angle you get when you look downward from a horizontal line of sight.
From your eyes, draw an imaginary horizontal line. The angle down from that line to the object is the angle of depression.
In the following drawing, the angle of depression is indicated by the symbol theta (θ).
Example 1
A 17-metre extension ladder rests against a vertical wall. The foot of the ladder is positioned 5 metres away from the wall on level ground. Determine, to the nearest tenth of a degree, the angle between the ladder and the ground. Include a clear sketch of the right triangle: wall (vertical), ground (horizontal), ladder (hypotenuse). Label all sides and place the symbol θ at the ground contact point to denote this angle.
Angle of inclination vs angle of depression: we work with the angle of inclination in this case since that's what is defined in the example.
Step 1: Draw a diagram to represent the situation and label it according to the information provided.
Remember that the ladder is leaning against the wall, so it is technically the hypotenuse of the triangle.
Step 2: Determine which side of the triangle is the opposite side, which side of the triangle is the adjacent side, and which side of the triangle is the hypotenuse.
Hypotenuse: the longest side, always across from the right angle.
Opposite: the side directly across from your chosen angle.
Adjacent: the side that touches your chosen angle, but is not the hypotenuse.
The hypotenuse is the length of the ladder (its length is 17 m).
The opposite side is the wall (its length is unknown).
The adjacent side is the length between the base of the ladder and the house (this length is 5 m).
Step 3: Determine which trigonometric ratio to use (sine, cosine, or tangent).
Sin(θ) = opposite / hypotenuse
Cos(θ) = adjacent / hypotenuse
Tan(θ) = opposite / adjacent
Notice that we know the adjacent side (5 m) and the hypotenuse (17 m). This means we need to use cosine.
We don't need sine or tangent because both of those require the opposite side, which is not required to answer the question since we already know the adjacent side and the hypotenuse and are looking for an angle. If you want to find an angle, you only need two sides.
Step 4: Set up the equation
Cos(θ) = 5 / 17
Step 5: Solve for the required angle
On your calculator, use the inverse cosine function to find the angle. It's usually found by pressing "2nd" or "shift" before pressing the cosine button on your calculator. It looks like cos⁻¹.
Make sure your calculator is in degree mode (DEG mode) and that you round your answer to the nearest tenth of a degree (one decimal place).
θ = cos⁻¹(5/17) = 72.9°
Therefore, the angle of inclination is 72.9°
Example 2
From the top of a 20-metre lighthouse, a lookout spots a buoy. The line of sight makes a 7° angle below the horizontal. How far from the lighthouse’s base, measured horizontally across the water, is the buoy? Give your answer to the nearest metre.
Angle of inclination vs angle of depression: we work with the angle of depression in this case since that's what is defined in the example.
Step 1: Draw a diagram to represent the situation and label it according to the information provided.
Remember that This is an angle you get when you look downward from a horizontal line of sight. I drew the horizontal line of sight as an imaginary dotted line.
Notice that the given angle of 7° ends up being outside the triangle of interest; I will call this angle θ₁ (shown in white). I then define an angle right next to it within the triangle of interest; I will call this angle θ₂ (shown in gold).
We would prefer to use θ₂ since it's actually inside of the triangle we're interested in. To find it, we use the fact that θ₁ and θ₂ must add up to 90°. This is always the case with two angles that share a corner such as θ₁ and θ₂.
θ₁ + θ₂ = 90°
θ₂ = 90° - θ₁
θ₂ = 90° - 7° = 83°
Therefore, we obtain the following complete triangle that best suits the context of the question. For simplicity, I will now refer to θ₂ as θ.
Step 2: Determine which side of the triangle is the opposite side, which side of the triangle is the adjacent side, and which side of the triangle is the hypotenuse.
Hypotenuse: the longest side, always across from the right angle.
Opposite: the side directly across from your chosen angle.
Adjacent: the side that touches your chosen angle, but is not the hypotenuse.
The hypotenuse is the line of sight from the sailor to the buoy (this length is unknown).
The opposite side is the distance from the buoy to the base of the lighthouse (we want to find this).
The adjacent side is the height of the lighthouse (this height is 20 m).
Step 3: Determine which trigonometric ratio to use (sine, cosine, or tangent).
Sin(θ) = opposite / hypotenuse
Cos(θ) = adjacent / hypotenuse
Tan(θ) = opposite / adjacent
Notice that we know the adjacent side (20 m) and we want to find the opposite side (the distance from the buoy to the base of the lighthouse. This means we need to use tangent.
We don't need sine or cosine because both of those require the hypotenuse, which is not required to answer the question since we already know the adjacent side, have the angle, and want the opposite side. If you know one side and an angle, you can find any other side.
Note that we know angle θ is 83°
Step 4: Set up the equation.
Tan(83°) = opposite / 20
Step 5: Solve for the required side.
Since the side we want is on the top of the fraction on the right-side, multiply both sides by the bottom of the fraction (20). Remember to round your answer to the nearest metre (no decimal place).
20 ⋅ Tan(83°) = opposite
20 ⋅ Tan(83°) = 163 m.
On your calculator, you probably got 162.8869286. Since we are told to round to the nearest metre, we round up to 163 m.
Therefore, the distance from the buoy to the base of the lighthouse is 163 m.
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