Angles of elevation and depression

When you look at something higher than your eyes, the angle of elevation ($\displaystyle \alpha$ in the drawing) is the angle between the horizontal line from your eyes and the line of sight. When you look at something lower than your eyes, the angle of depression ($\displaystyle \beta$) is the angle between the horizontal line and the line of sight down.

• Both angles are measured from the horizontal line, not from the vertical one.
• The horizontal line, the line of sight and the vertical line make a right-angled triangle, and that is where we calculate.

Gal stands 40 m from a building, her eyes 1.6 m above the ground.
The angle of elevation to the top of the building is $\displaystyle 35^\circ$.
How tall is the building?

The solution below finds the height $\displaystyle x$ above the eyes, and adds the height of the eyes to it.

• $\displaystyle x$ is opposite the angle, and 40 next to it: $\displaystyle {\tan35^\circ}={\frac{x}{40}}$, so $\displaystyle {x}={40\cdot\tan35^\circ\approx28.01}$ m.
• The building’s height: $\displaystyle {1.6+28.01}={29.61}$ m.

Yoav stands 50 m from a water tower, his eyes 1.7 m above the ground.
The angle of elevation to the top of the tower is $\displaystyle 28^\circ$.
How tall is the tower, in metres? Round to two decimal places.

From the top of a lighthouse 45 m tall, a boat is seen at an angle of depression of $\displaystyle 12^\circ$.
How far is the boat, $\displaystyle d$, from the foot of the lighthouse?

• The horizontal line at the top of the lighthouse is parallel to the ground, so the angle of depression equals the angle between the ground and the line of sight at the boat (alternate angles).
• At the boat: the height 45 is opposite the angle, and $\displaystyle d$ next to it. $\displaystyle {\tan12^\circ}={\frac{45}{d}}$.
• $\displaystyle {d}={\frac{45}{\tan12^\circ}\approx211.71}$ m.

From a balcony 20 m up, a parked car is seen at an angle of depression of $\displaystyle 25^\circ$.
How far is the car from the building, in metres? Round to two decimal places.

A kite’s string is 60 m long, taut and straight, at an angle of elevation of $\displaystyle 40^\circ$.
The string is held 1.2 m above the ground.
How high is the kite, in metres? Round to two decimal places.

Tamar stands at the point $\displaystyle A$ on a river bank, opposite a tree $\displaystyle T$ on the other bank.
She walks 30 m along the bank, to the point $\displaystyle B$.
From $\displaystyle B$, the angle between the bank and the line of sight to the tree is $\displaystyle 52^\circ$.
How wide is the river, $\displaystyle x$, in metres? Round to two decimal places.

A plane 300 m up comes down to land in a straight line, at an angle of $\displaystyle 3^\circ$ to the ground.
How many metres before the runway is it, measured along the ground? Round to two decimal places.