Angles of Elevation and Depression

A person stands $20$ $m$ from the base of a tower. The angle of elevation to the top of the tower is $35$°. Find the height of the tower.

• 1This forms a right-angled triangle: adjacent = $20$ (distance along ground), opposite = height (unknown), angle = $35$°.
• 2Use tan: tan($35$°) = height ÷ $20$
• 3Rearrange: height = $20$ × tan($35$°) ≈ $20\times 0.700$ ≈ $14.0$

Answer: ≈ $14.0$ $m$

From the top of a $50$ $m$ cliff, the angle of depression to a boat is $25$°. Find the distance from the base of the cliff to the boat.

• 1The angle of depression from the cliff top equals the angle of elevation from the boat (alternate angles) — both are $25$°.
• 2This forms a right-angled triangle: opposite = $50$ (height), adjacent = distance (unknown), angle = $25$°.
• 3Use tan: tan($25$°) = $50$ ÷ distance
• 4Rearrange: distance = $50$ / tan($25$°) ≈ $50$ / $0.4663$ ≈ $107.2$

Answer: ≈ $107.2$ $m$

Sarah stands $40$ $m$ from a tree. The angle of elevation from her eyes to the top is $45$°. Her eyes are $1.5$ $m$ above the ground. How tall is the tree? (Use $\tan 45^{\circ }=1$.)

• 1From eye level, opposite = adjacent × tan($45$°) = $40\times 1=40$ $m$.
• 2That $40$ $m$ is only the height above her eyes.
• 3Total tree height = $40+1.5$ = $41.5$ $m$.

Never forget to add (or subtract) the observer's eye height when the question gives it.