A boat is 530 metres away from the base of a 400-metre-high cliff. Find the angle of depression of the boat from the top of the cliff, giving your answer in radians to three decimal places.
Let’s begin by sketching a diagram of this scenario. Remember, a sketch does not need to be to scale, but it should be roughly in proportion so you can check the suitability of any answers you get.
The cliff is 400 metres high and the boat is 530 metres away from the base of it. We assume that the angle between the cliff and the water is 90 degrees. The line of sight forms the third side of the triangle.
We’re being asked to find the angle of depression of the boat from the top of the cliff. If a person stands and looks down at an object, the angle of depression is the angle between the horizontal line of sight and the object. In this case, that’s the angle 𝜃.
Whilst this angle isn’t in our triangle, we can use the fact that the horizontal line of sight and the sea are parallel lines. Alternate angles are equal, so we can show that this angle is also 𝜃.
Now we have a right-angled triangle with two known lengths in which we’re trying to find the size of one of the angles. We need to use right angle trigonometry to do this. Let’s label the sides of the triangle.
The hypotenuse is the longest side of this triangle. It’s the side situated directly opposite the right angle. The opposite side is the side opposite the given angle. It’s the one furthest away from the angle 𝜃 within our triangle. Finally, the adjacent side is the other side. It’s located next to the angle 𝜃.
We can see that we know both the length of the opposite and the adjacent sides. This means we need to use the tan ratio. Tan of 𝜃 is equal to opposite over adjacent. We can substitute the numbers from our triangle into the formula, giving us tan of 𝜃 is equal to 400 over 530.
Then we solve this equation by finding the inverse tan of both sides. The inverse tan of tan 𝜃 is simply 𝜃, so 𝜃 is equal to the inverse tan of 400 over 530, which is 0.64651. Correct to three decimal places, 𝜃 is equal to 0.647 radians. The angle of depression of the boat from the top of the cliff is 0.647 radians.