Trigonometry: Height & Distance – Angles, Ratios & Real-World Problems

TAKEAWAYS:

• Learn the three primary trigonometric ratios (SOH-CAH-TOA) and how they relate to right-angled triangles.
• Understand angles of elevation (looking up) and angles of depression (looking down) in real-life scenarios.
• Solve problems using height-and-distance formulas without needing direct measurements.
• Apply trigonometry to shadow problems, building heights, and navigation (e.g., finding distances across rivers).
• Master word-to-diagram conversion—drawing accurate sketches is half the battle in SEE exams.
• Avoid common mistakes like misidentifying the adjacent/hypotenuse side or confusing elevation/depression angles.

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1. Introduction to Trigonometry in Height & Distance

Trigonometry is the study of triangles, especially right-angled triangles. In this unit, we use trigonometric ratios to find unknown heights or distances when direct measurement is difficult or impossible.

Key Terms:

• Angle of Elevation: The angle between the horizontal and the line of sight upwards (e.g., looking at the top of a tower).
• Angle of Depression: The angle between the horizontal and the line of sight downwards (e.g., looking at an object below eye level).
• Line of Sight: The straight line from the observer’s eye to the object.

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2. The Three Primary Trigonometric Ratios

For any right-angled triangle, the ratios of the sides are constant for a given angle. These are called sine (sin), cosine (cos), and tangent (tan).

Remember: SOH-CAH-TOA (Say "Oh, Come On, Take Off All Your Clothes!" to remember).

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3. Solving Problems Using Trigonometry

Step-by-Step Approach:

1. Draw a diagram: Sketch the scenario (even if not given).
2. Label the right angle: Ensure it’s a right-angled triangle.
3. Identify the given angle (θ) and the sides (Opposite, Adjacent, Hypotenuse).
4. Choose the correct ratio (SOH, CAH, or TOA).
5. Rearrange the formula to solve for the unknown.
6. Calculate and give the answer in the required units.

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Example 1: Finding the Height of a Tower

Problem: A man stands 50 meters away from a tower. The angle of elevation from his eye level to the top of the tower is 30°. If his eyes are 1.5 meters above the ground, find the height of the tower.

Solution:

1. Draw the diagram:

   • Let the height of the tower above the man’s eye level be $h$.
   • Total height of the tower = $h+1.5$ meters.

2. Identify the sides:

   • Opposite side to angle θ = $h$ (height above eye level).
   • Adjacent side = 50 meters (distance from the tower).
   • Angle θ = 30°.

3. Choose the ratio: Since we have the opposite and adjacent sides, use tan θ. $$\tan 30°=\frac{\text{Opposite}}{\text{Adjacent}}=\frac{h}{50}$$

4. Rearrange and solve: $$h=50\times \tan 30°$$ We know $\tan 30°=\frac{1}{\sqrt{3}}\approx 0.577$. $$h=50\times 0.577=28.87\text{ meters}$$

5. Total height of the tower: $$\text{Total height}=h+1.5=28.87+1.5=30.37\text{ meters}$$ Answer: The height of the tower is 30.37 meters.

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Example 2: Angle of Depression

Problem: From the top of a 20-meter-high building, the angle of depression to the base of another building is 45°. If the two buildings are 30 meters apart, find the height of the second building.

Solution:

1. Draw the diagram:

   • Let the height of the second building be $x$ meters.
   • The horizontal distance between the buildings is 30 meters.
   • The vertical difference is $20-x$ meters.

2. Identify the sides:

   • Opposite side = $20-x$ (difference in height).
   • Adjacent side = 30 meters (distance between buildings).
   • Angle of depression = 45° (angle of elevation from the second building to the first is also 45°).

3. Choose the ratio: Use tan θ because we have opposite and adjacent sides. $$\tan 45°=\frac{20-x}{30}$$ We know $\tan 45°=1$, so: $$1=\frac{20-x}{30}$$

4. Solve for $x$: $$\begin{gathered} 20-x=30 \\ x=20-30=-10 \end{gathered}$$ Wait! This doesn’t make sense (negative height). Let’s re-examine the diagram.

   Correction: The angle of depression is from the top of the first building to the base of the second building. The correct opposite side is the height of the second building itself if we consider the line of sight horizontally. Redraw the diagram properly:

   • The horizontal distance is 30 meters.
   • The vertical drop is 20 meters (height of the first building).
   • The angle of depression is 45°, so the angle of elevation from the base of the second building to the top of the first is also 45°.

   Now, the opposite side is the height of the second building ($x$), and the adjacent side is the horizontal distance (30 meters). $$\begin{gathered} \tan 45°=\frac{x}{30} \\ 1=\frac{x}{30} \\ x=30\text{ meters} \end{gathered}$$ Answer: The height of the second building is 30 meters.

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4. Shadow Problems

Shadow problems involve the angle of elevation of the sun and the height of an object casting a shadow.

Example 3: Finding the Height of a Pole Using Shadow

Problem: A pole casts a shadow of 10 meters when the angle of elevation of the sun is 60°. Find the height of the pole.

Solution:

1. Draw the diagram:

   • Opposite side (height of pole) = $h$.
   • Adjacent side (shadow) = 10 meters.
   • Angle θ = 60°.

2. Choose the ratio: Use tan θ: $$\tan 60°=\frac{h}{10}$$ We know $\tan 60°=\sqrt{3}\approx 1.732$.

3. Solve for $h$: $$h=10\times \tan 60°=10\times 1.732=17.32\text{ meters}$$ Answer: The height of the pole is 17.32 meters.

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5. Comparing Trigonometric Ratios

Here’s a quick reference table for common angles (0°, 30°, 45°, 60°, 90°):

Note: Memorize these values—they appear frequently in SEE exams!

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6. Real-Life Applications of Trigonometry

Trigonometry is used in many fields:

• Architecture: Calculating heights of buildings, slopes of roofs.
• Navigation: Determining distances between ships or planes.
• Astronomy: Measuring distances to stars or planets.
• Surveying: Mapping land and measuring heights.
• Engineering: Designing bridges, roads, and structures.

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7. Common Mistakes to Avoid

1. Misidentifying sides: Always double-check which side is opposite, adjacent, or hypotenuse.
2. Confusing elevation/depression: Elevation is upwards, depression is downwards.
3. Forgetting units: Always include meters, centimeters, or kilometers in your answer.
4. Incorrect angle placement: Ensure the angle is correctly placed in the diagram.
5. Rounding errors: Keep intermediate steps precise before rounding the final answer.

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8. SEE-Style Practice Questions

Short Answer Questions (1 mark each)

1. Define angle of elevation.
2. If $\tan \theta =\frac{3}{4}$, what is $\sin \theta$?
3. In a right-angled triangle, if the hypotenuse is 10 cm and the angle is 30°, find the opposite side.

Long Answer Questions (5-7 marks)

4. A tree casts a shadow of 15 meters when the angle of elevation of the sun is 45°. Find the height of the tree.
5. From the top of a 50-meter-high cliff, the angle of depression to a boat is 30°. How far is the boat from the base of the cliff?
6. A ladder leans against a wall at an angle of 60° to the ground. If the ladder is 10 meters long, how high up the wall does it reach?
7. Two buildings are 40 meters apart. The angle of elevation from the top of the shorter building (20 meters) to the top of the taller building is 30°. Find the height of the taller building.

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Exam Tip

1. Always draw a diagram: Even if the problem doesn’t provide one, sketching helps visualize the scenario.
2. Label all given and unknown quantities: Clearly mark the angle, sides, and what you need to find.
3. Use the correct trigonometric ratio: SOH-CAH-TOA is your best friend—pick the right one!
4. Show all steps: Partial credit is given for correct steps, even if the final answer is wrong.
5. Check units: Ensure your answer is in the correct unit (meters, centimeters, etc.).
6. Practice with real-world problems: Trigonometry in SEE often involves heights of trees, buildings, or angles of shadows—practice these!
7. Memorize key values: Know the sine, cosine, and tangent of 30°, 45°, and 60° by heart.

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Summary

• Trigonometry helps us find unknown heights and distances using angles.
• The three key ratios are sin θ, cos θ, tan θ (SOH-CAH-TOA).
• Angle of elevation is upwards; angle of depression is downwards.
• Always draw a diagram, label sides, and choose the right ratio.
• Practice shadow problems, building heights, and navigation scenarios for SEE success!

Good luck with your exams! 🚀