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AI SL · 12.10

Geometry & Trigonometry

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12.10.7

Angles of elevation and depression

12.10.7

Angles of elevation and depression

Target: I can say whether an angle is an angle of elevation or of depression, and mark it from the horizontal. (+3 more)

By the end of this section

Key points

Elevation and depression

  • Both angles are measured from the horizontal line through the observer's eyes.
  • Elevation: looking up, above the horizontal. Depression: looking down, below the horizontal.
  • The angle of depression from A to B equals the angle of elevation from B to A. They are alternate angles.

Two sightings in a line

  • Draw the triangle between the two observation points and the top of the object.
  • The angle at the nearer point inside the triangle is 180° minus the nearer angle of elevation. The angle at the top is the difference of the two elevations.
  • Use the sine rule to find the line of sight from the nearer point. Then use a right-angled triangle for the height.
  • Add the observer's eye height at the end if the height is measured from eye level.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
One angle of elevation

Ana stands 45 m from the foot of a tree. The angle of elevation of the top of the tree from her eyes is 27°. Her eyes are 1.5 m above the ground. Find the height of the tree.

45 mh27°AT
Working
  1. The angle of elevation

    Measured from the horizontal, upwards

    The horizontal line from her eyes and the line of sight make the angle. The dashed horizontal is 45 m.

  2. Height above her eyes

    \(\tan 27^\circ = \dfrac{h}{45}\), \(h = 45\tan 27^\circ = 22.93\)

    The right-angled triangle has the 45 m adjacent and h opposite.

  3. Add her eye height

    22.93 + 1.5 = 24.4 m

    h stops at eye level, not at the ground.

Video walkthrough coming
Two angles of elevation

Sam looks at the top of a building. The angle of elevation is 38°. He walks 40 m directly away from the building and the angle of elevation is now 24°. His eyes are 1.6 m above the ground. Find the height of the building.

40 mh24°38°BAT
Working
  1. 40 m24°38°BAT
    Angles in triangle ABT

    \(\angle ABT = 24^\circ\), \(\angle BAT = 180^\circ - 38^\circ = 142^\circ\), \(\angle ATB = 38^\circ - 24^\circ = 14^\circ\)

    The angle at A inside the triangle is the supplement of 38°. The angle at T comes from the angle sum, or from the exterior angle.

  2. Sine rule for AT

    \(\dfrac{AT}{\sin 24^\circ} = \dfrac{40}{\sin 14^\circ}\), so \(AT = 67.25\)

    AT is the side shared by both triangles. You know d and the angle opposite it.

  3. Right-angled triangle

    \(h = AT \sin 38^\circ = 67.25 \times \sin 38^\circ = 41.40\)

    Now use the right-angled triangle at A with AT as hypotenuse.

  4. Eye height

    Add the eye height: 41.40 + 1.6 = 43.0

    Add the height of Sam's eyes above the ground.

Video walkthrough coming
Angles of depression

A ranger on a headland looks at a boat. The angle of depression of the boat is 12°. The boat sails 300 m directly towards the headland and the angle of depression is then 20°. Find the height of the ranger above the sea.

300 mh12°20°BAT
Working
  1. 300 m12°20°BAT
    Depression becomes elevation

    Angle of depression = angle of elevation from the boat

    The horizontal at the ranger and the sea are parallel, so the angles are alternate. Treat 12° and 20° as elevations from the boats.

  2. Triangle angles

    \(\angle ABT = 12^\circ\), \(\angle BAT = 180^\circ - 20^\circ = 160^\circ\), \(\angle ATB = 20^\circ - 12^\circ = 8^\circ\)

    B is the first boat position and A is the second.

  3. Sine rule

    \(\dfrac{AT}{\sin 12^\circ} = \dfrac{300}{\sin 8^\circ}\), so \(AT = 448.2\)

    Find the shared side AT.

  4. Right-angled triangle

    \(h = AT \sin 20^\circ = 448.2 \times \sin 20^\circ = 153.3\)

    The ranger's height above the sea is h. There is no eye height to add: the angles are measured from the ranger's position.

Video walkthrough coming