Heights and Distances: Angles of Elevation and Depression

Concepts and Skills: Heights and Distances

• Apply knowledge of solving right-angled triangles to problems involving HEIGHTS, DISTANCES and BEARING.

• Represent a real-life problem as a right-angled triangle or triangles.

• Required resources mentioned: paper-based resources (textbook Trigonometry section) and digital resources (online videos).

• NB! Bearing also involves Trigonometry; watch the bearing video link provided.

Labeling and Key Ratios

• You should label the three sides of a triangle in terms of:

  • hypotenuse (the side opposite the 90° angle),

  • opposite (the side opposite the angle of interest),

  • adjacent (the side next to the angle of interest, not the hypotenuse).

• Trigonometric ratios to know:

  • $\sin\theta = \frac{\text{opp}}{\text{hyp}}$

  • $\cos\theta = \frac{\text{adj}}{\text{hyp}}$

  • $\tan\theta = \frac{\text{opp}}{\text{adj}}$

• Mnemonic: SOH CAH TOA for remembering the ratios.

• Remember: Bearings also involve trigonometry and can be analyzed with these ratios when converting directions into components or angles.

Angles of Elevation and Depression (definitions)

• Angle of Elevation:

  • The angle from the horizontal line upward to an object.

  • In a diagram, the sight line would be above the horizontal line.

• Angle of Depression:

  • The angle from the horizontal line downward to an object.

  • In a diagram, the sight line would be below the horizontal line.

• Both concepts use right-angled triangles and trigonometric ratios to relate heights and distances.

Example 1: Distance on the ground between a building and a hotel (Angle of Depression)

• Given: A building is 200 m high. The observer at the top looks down to the base of a hotel; the angle of depression is 57°.

• Let y be the distance on the ground between the building and the hotel.

• Relationship:

  • $\tan(57^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{200}{y}$

• Solve for y:

  • $y = \frac{200}{\tan(57^{\circ})}$

• Numeric approximation:

  • $y \approx \frac{200}{\tan(57^{\circ})} \approx 129.9\ \text{m}$

• Sketch/Cognitive steps described in the material:

  • Draw the right-angled triangle with the vertical height of the building (200 m) as the opposite side to the angle of depression.

  • Mark the right angle between the building and the ground.

  • The horizontal line through the observer is parallel to the ground; by alternate interior angles, the angle inside the triangle can be marked as 57°.

  • The distance on the ground between the building and the hotel is the adjacent side y.

• Practical takeaway: This demonstrates how height and horizontal distance relate to an angle of depression using tangent.

Example 2: Height of the hotel (Angle of Elevation)

• Setup: From the top of the building (point A), the angle of elevation to the top of the hotel (point C) is 57°. The horizontal ground distance between the buildings is y (as found in Example 1).

• Let h be the height of the hotel.

• Relationship using the angle of elevation:

  • The height difference from the top of the building to the top of the hotel is (h − 200).

  • $\tan(57^{\circ}) = \frac{h - 200}{y}$

• Solve for h:

  • $h = 200 + y\,\tan(57^{\circ})$

• Substituting the value of y from Example 1:

  • Since $y = \dfrac{200}{\tan(57^{\circ})}$,

  • $h = 200 + \left(\dfrac{200}{\tan(57^{\circ})}\right) \tan(57^{\circ}) = 200 + 200 = 400\ \text{m}$

• Conclusion:

  • Height of the hotel, $h \approx 400\ \text{m}$

• Key idea: Using the same angle of elevation and the same base distance, the height difference can be related back to the original height via the tangent function; the two-step process demonstrates consistency in right-triangle relations.

Practical workflow for HEIGHTS and DISTANCES problems

• Step 1: Label the triangle(s) with opposite, adjacent, and hypotenuse relative to the angle of interest.

• Step 2: Decide whether you are dealing with angle of elevation or angle of depression.

• Step 3: Write the appropriate tangent relationship (or sine/cosine if the hypotenuse is involved) using the known quantities.

• Step 4: Solve for the unknown (distance, height, or height difference).

• Step 5: If multiple steps are involved (e.g., first distance, then height), substitute results from earlier steps as needed.

• Step 6: Check units and reasonableness of the answer (e.g., heights should be positive, distances should be non-negative).

Connections to bearings and real-world relevance

• Bearings involve angular measurements in navigation and map reading; trigonometric relationships are used to convert bearings into components or distances.

• Historical context: Trigonometry originated to solve practical problems in construction and navigation, illustrating its real-world relevance.

• The techniques shown here enable solving problems about visibility, line of sight, and distance estimation from observation points.

Resources mentioned (for reference)

• Paper-based: Consult your Trigonometry section in the textbook.

• Digital resources:

  • Intro to Heights and Distances (Khan Academy): https://www.youtube.com/watch?v=TgQs7k5p2Ag

  • Exact trigonometric values using hand trick - Special Angles: https://www.youtube.com/watch?v=TyrM8G1Mqil

  • Angles of Elevation and Depression: https://www.youtube.com/watch?v=70Nj6TKvi2g

  • NB! Bearing video: https://www.youtube.com/watch?v=Udpkd_7Eji

Quick recap: Key formulas to memorize

• $\sin\theta = \frac{\text{opp}}{\text{hyp}}$

• $\cos\theta = \frac{\text{adj}}{\text{hyp}}$

• $\tan\theta = \frac{\text{opp}}{\text{adj}}$

• Angle of Elevation: angle from horizontal up to an object.

• Angle of Depression: angle from horizontal down to an object.

• In problems like Example 1 and Example 2, the tangent relationship is the primary tool to connect heights and distances through horizontal distances and height differences.