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Core + Extended Trigonometry

Angles of Elevation and Depression for IGCSE Maths

Solving problems involving angles measured above and below the horizontal. This subtopic is part of Trigonometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Unders

What You Need to Know

Solving problems involving angles measured above and below the horizontal. This subtopic is part of Trigonometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding angles of elevation and depression is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Angles of Elevation and Depression

The angle of elevation is the angle measured upward from the horizontal to a point above. The angle of depression is the angle measured downward from the horizontal to a point below. Both angles are always measured from the horizontal, not from a vertical line. In IGCSE 0580, these appear with right-angled triangle trigonometry to find heights of buildings, trees, cliffs, or distances across valleys. Alternate angles make the angle of elevation from A to B equal to the angle of depression from B to A.

Step-by-Step Method

  1. 1

    Draw the right-angled triangle

    Identify the horizontal distance, the vertical height, and the hypotenuse (line of sight). Label the angle of elevation or depression clearly.

  2. 2

    Identify the angle position

    The angle of elevation/depression is at the observer's eye level. It is between the horizontal and the line of sight.

  3. 3

    Label Opposite, Adjacent, Hypotenuse

    Relative to the angle: Opposite = the vertical height, Adjacent = the horizontal distance, Hypotenuse = the line of sight (slant distance).

  4. 4

    Choose the trigonometric ratio

    O and H → sin. A and H → cos. O and A → tan. In most elevation/depression problems, O and A are given/required, so tan is used most often.

  5. 5

    Solve and interpret

    Calculate and round to the required accuracy. Check that the answer is sensible: a height cannot be negative and should be physically reasonable.

Worked Example

Question

From a point P on horizontal ground, the angle of elevation of the top T of a vertical tower is 38°. Point P is 45 m from the base B of the tower. Find the height of the tower, correct to 3 significant figures.

Solution

Step 1: Draw the right-angled triangle PBT. PB = 45 m (horizontal), BT = height (vertical), angle at P = 38°. Step 2: Relative to angle P: BT is Opposite, PB is Adjacent. Use tan. tan 38° = BT / 45 Step 3: Solve. BT = 45 × tan 38° BT = 45 × 0.7813 BT = 35.157... m Answer: Height of tower = 35.2 m (3 s.f.)

Exam Tips for Angles of Elevation and Depression

  • The angle of elevation is always measured from the HORIZONTAL upward — never from the vertical.
  • Angle of depression from X to Y = angle of elevation from Y to X (alternate angles on parallel horizontal lines).
  • In most problems, you know the horizontal distance and need the height, so tan is the most-used ratio for elevation/depression.
  • For two-observer problems (angles from two different points), set up two equations in two unknowns (height h and one horizontal distance).

Practice Questions

Q1: From the top of a cliff 80 m high, the angle of depression of a boat is 22°. Find the horizontal distance from the boat to the base of the cliff.

Show hint

The angle of depression from the cliff top equals the angle of elevation from the boat (alternate angles). Opposite = 80 m, find Adjacent using tan 22°.

Q2: A flagpole is 12 m tall. Find the angle of elevation from a point 20 m away from the base of the flagpole on level ground.

Show hint

tan θ = 12/20. Use tan⁻¹ to find θ.

Q3: From two points A and B, 50 m apart on level ground, the angles of elevation of a point C directly above AB are 42° and 28° respectively. Find the height of C above the ground.

Show hint

If the height is h and horizontal distance from A is x, then tan 42° = h/x and tan 28° = h/(50-x). Solve simultaneously.

Frequently Asked Questions

What is angles of elevation and depression in IGCSE Maths?

Solving problems involving angles measured above and below the horizontal.

Is angles of elevation and depression in the Core or Extended syllabus?

Angles of Elevation and Depression is part of the Core and Extended syllabus for IGCSE Mathematics 0580.

How do I revise angles of elevation and depression effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on angles of elevation and depression rather than trying to cover everything at once.

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