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Trigonometry: angles of elevation and depression — KCSE Mathematics

KCSE Mathematics · 108 practice questions · 3 syllabus objectives · 3 revision lessons

38 easy37 medium33 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define and distinguish between the angle of elevation and angle of depression and represent them in a diagram

Apply trigonometric ratios to solve problems involving angles of elevation, depression and bearings

Trigonometry: angles of elevation and depression

Revision Notes

Concise lesson notes for Trigonometry: angles of elevation and depression, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Angles of Elevation and Depression

In trigonometry, the angle of elevation is formed when you look upwards from a horizontal line to an object, while the angle of depression is formed when you look downwards from a horizontal line to an object.

To visualize this:

  • Draw a horizontal line representing the observer's line of sight.
  • The angle of elevation is the angle between the horizontal line and the line of sight to an object above.
  • The angle of depression is the angle between the horizontal line and the line of sight to an object below.

Diagram Representation:

  1. Draw a horizontal line (the observer's sight).
  2. Draw an upward line to represent the object above, marking the angle of elevation (θ).
  3. Draw a downward line to represent the object below, marking the angle of depression (φ).

These angles are crucial in solving problems related to heights and distances in real-life scenarios.

Key points to remember

  • Angle of elevation is above the horizontal line.
  • Angle of depression is below the horizontal line.
  • Both angles are measured from the horizontal line.
  • They are formed with respect to an observer's line of sight.
  • Use diagrams to illustrate these concepts clearly.

Worked example

Define the angle of elevation and angle of depression.

  • Angle of elevation: the angle formed when looking up from horizontal.
  • Angle of depression: the angle formed when looking down from horizontal.

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Lesson 2: Using Trigonometry for Angles of Elevation and Depression

Objective: Apply trigonometric ratios to solve problems involving angles of elevation, depression and bearings

In trigonometry, angles of elevation and angles of depression are essential concepts. The angle of elevation is formed when you look up at an object, while the angle of depression is formed when you look down at an object.

To solve problems involving these angles, we use the basic trigonometric ratios:

  • Sine (sin) = Opposite / Hypotenuse
  • Cosine (cos) = Adjacent / Hypotenuse
  • Tangent (tan) = Opposite / Adjacent

Example Problem: A person is standing 50 meters away from a tree. The angle of elevation to the top of the tree is 30 degrees. Find the height of the tree.

Solution:

  1. Let the height of the tree be h.
  2. Use the tangent ratio:
    • tan(30°) = h / 50
  3. Rearranging gives:
    • h = 50 * tan(30°)
  4. Calculate:
    • h = 50 * (1/√3) = 50/√3 ≈ 28.87 meters.

Thus, the height of the tree is approximately 28.87 meters.

  • Angle of elevation is the angle looking up.
  • Angle of depression is the angle looking down.
  • Use tan, sin, and cos for calculations.
  • Identify opposite, adjacent, and hypotenuse sides.
  • Apply correct trigonometric ratios for accurate results.

A tower casts a shadow 40 meters long when the angle of elevation of the sun is 45 degrees. Find the height of the tower.

Solution:

  1. Let the height of the tower be h.
  2. Using tan: tan(45°) = h / 40.
  3. h = 40 * tan(45°) = 40 * 1 = 40 meters.
Lesson 3: Understanding Angles of Elevation and Depression

Objective: Trigonometry: angles of elevation and depression

In trigonometry, the angle of elevation is the angle formed when you look up from a horizontal line to an object above you. Conversely, the angle of depression is the angle formed when you look down from a horizontal line to an object below you.

To solve problems involving these angles, we often use right-angled triangles. Here are key steps to remember:

  • Identify the horizontal line from your viewpoint.
  • Draw the angle of elevation or depression from this line.
  • Label the opposite side (height or depth) and the adjacent side (distance from the object).

Use trigonometric ratios such as tan(θ) = opposite/adjacent to find unknown lengths or angles.

For example, if you stand 30 meters from a building and the angle of elevation to the top is 60 degrees, you can find the height of the building using:

  • tan(60°) = height/30m.
  • Therefore, height = 30m * tan(60°) = 30m * √3 ≈ 51.96m.
  • Angle of elevation looks up from a horizontal line.
  • Angle of depression looks down from a horizontal line.
  • Use right-angled triangles to solve problems.
  • tan(θ) = opposite side / adjacent side.
  • Identify and label all sides before calculations.

A person stands 40m from a tree and sees the top at an angle of elevation of 45 degrees. Find the height of the tree.

  • tan(45°) = height/40m.
  • Therefore, height = 40m * tan(45°) = 40m * 1 = 40m.

Sample Questions

Read 3 questions and answers free. Sign up to access all 108 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER2 marks

From a point on the ground, the angle of elevation to the top of a tree is 30°. If the distance from the point to the base of the tree is 10 m, calculate the height of the tree. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Use the formula h = d × tan(30°) (1 mk)
Height of the tree = 10 × (1/√3) = (10√3)/3 m (1 mk)
2
easySHORT ANSWER4 marks

Explain the difference between the angle of elevation and the angle of depression and describe their significance in real-life situations. (4 marks)

Answer & marking scheme

Part (a) — 2 marks
The angle of elevation is measured upwards from the horizontal (1 mk)
The angle of depression is measured downwards from the horizontal (1 mk)
Part (b) — 2 marks
Angle of elevation is used by surveyors to measure the height of mountains or buildings (1 mk)
Angle of depression is used by pilots to determine the height of an aircraft above the ground (1 mk)
3
easySHORT ANSWER4 marks

Define the angle of elevation and the angle of depression. Provide one practical example for each. (4 marks)

Answer & marking scheme

Part (a) — 2 marks
The angle of elevation is formed between the horizontal line and the line of sight when looking upwards at an object above the horizontal (1 mk)
Example: Looking up at a tall building from the ground (1 mk)
Part (b) — 2 marks
The angle of depression is formed between the horizontal line and the line of sight when looking downwards at an object below the horizontal (1 mk)
Example: Looking down from a balcony at people on the street below (1 mk)
4

From a point on the ground, the angle of depression to a car parked on the road is 45°. If the observer's height is 10 m, determine the horizontal distance from the base of the observer to the car. (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Trigonometry: angles of elevation and depression" cover?

Trigonometry: angles of elevation and depression covers Define and distinguish between the angle of elevation and angle of depression and represent them in a diagram; Apply trigonometric ratios to solve problems involving angles of elevation, depression and bearings; Trigonometry: angles of elevation and depression, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Trigonometry: angles of elevation and depression?

HighMarks has 108 Trigonometry: angles of elevation and depression practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Trigonometry: angles of elevation and depression for the KCSE exam?

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