Angles of elevation and depression questions with answers

Questions

1. Question 1Find a height

   From a point $30$ m from the foot of a flagpole on level ground, the angle of elevation of the top is $23^\circ$. Find the height of the flagpole, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Which ratio links h, the distance and the angle?$\tan 23^\circ = \dfrac{h}{30}$
   2. Step 2: Make h the subject$h = 30 \tan 23^\circ$$h = 12.7342\ldots$$\phantom{h} = 12.7$ m (3 s.f.)
2. Question 2Angle of depression: find a distance

   From the top of a tower $40$ m high, the angle of depression of a buoy is $49^\circ$. Find the distance of the buoy from the foot of the tower, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Mark the angle at BAngle $TBF = 49^\circ$ (alt. ∠s)
   2. Step 2: Which ratio links the height, x and the angle at B?$\tan 49^\circ = \dfrac{40}{x}$
   3. Step 3: Make x the subject$x = \dfrac{40}{\tan 49^\circ}$$x = 34.7714\ldots$$\phantom{x} = 34.8$ m (3 s.f.)
3. Question 3Find the angle of elevation

   A tree $12$ m tall stands on level ground. Find the angle of elevation of its top from a point $40$ m from its foot, correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Which ratio links the two lengths and θ?$\tan \theta = \dfrac{12}{40}$
   2. Step 2: Use the inverse ratio$\theta = \tan^{-1}\left(\dfrac{12}{40}\right)$$\theta = 16.6992\ldots^\circ$$\phantom{\theta} = 16.7^\circ$ (1 d.p.)
4. Question 4Find the angle of depression

   A car is $40$ m from the foot of a lighthouse $45$ m high. Find the angle of depression of the car from the top of the lighthouse, correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Mark the angle at BAngle $TBF = \theta$ (alt. ∠s)
   2. Step 2: Which ratio links the two lengths and θ?$\tan \theta = \dfrac{45}{40}$
   3. Step 3: Use the inverse ratio$\theta = \tan^{-1}\left(\dfrac{45}{40}\right)$$\theta = 48.3664\ldots^\circ$$\phantom{\theta} = 48.4^\circ$ (1 d.p.)
5. Question 5Two angles of elevation

   $A$ and $B$ are points on level ground in a straight line with the foot $F$ of a lamp post, with $AB = 30$ m. The angles of elevation of the top $T$ from $A$ and $B$ are $32^\circ$ and $55^\circ$ respectively. Find the height of the lamp post, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Let BF = x: which ratio links h and the ground distance in each triangle?In triangle $BFT$: $h = x \tan 55^\circ$In triangle $AFT$: $h = (x + 30) \tan 32^\circ$
   2. Step 2: Make them equal and collect the x terms$x \tan 55^\circ = (x + 30) \tan 32^\circ$$x(\tan 55^\circ - \tan 32^\circ) = 30 \tan 32^\circ$
   3. Step 3: Find x$x = \dfrac{30 \tan 32^\circ}{\tan 55^\circ - \tan 32^\circ}$$x = 23.3369\ldots$
   4. Step 4: Then h$h = 23.3369\ldots \times \tan 55^\circ$$h = 33.3286\ldots$$\phantom{h} = 33.3$ m (3 s.f.)