Some Applications of Trigonometry: Class 10 Maths Chapter 9 Notes

Key Terms You Must Know

Line of Sight

The imaginary straight line drawn from the eye of an observer to the point on the object being viewed.

Horizontal Line

The straight line drawn parallel to the Earth's surface from the observer's eye.

Angle of Elevation

The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level. We raise our head to see the object.

Angle of Depression

The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level. We lower our head to see the object.

Trigonometric Ratios

The ratios of the sides of a right-angled triangle with respect to its acute angles. The main ratios are sine (sin), cosine (cos), and tangent (tan).

SOH CAH TOA

A mnemonic to remember the primary trigonometric ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

Must Remember for Problem Solving

• {"point":"The Angle of Elevation from point A to B is equal to the Angle of Depression from point B to A (Alternate Interior Angles)."}
• {"point":"Always draw a simple, labeled diagram first to visualize the problem."}
• {"point":"Assume the object (tower, building, tree) is perpendicular to the ground, forming a 90° angle."}
• {"point":"Memorize the trigonometric table values for 0°, 30°, 45°, 60°, and 90°, especially for sin, cos, and tan."}
• {"point":"Choose the correct trigonometric ratio (sin, cos, tan) based on what sides are given and what needs to be found."}
• {"point":"If the observer's height is given, it must be accounted for in the diagram and calculations."}
• {"point":"Tan ratio is most commonly used as problems often involve opposite (height) and adjacent (distance) sides."}
• {"point":"Ensure all units of length (meters, cm) are consistent throughout the problem."}
• {"point":"For problems with two triangles, identify the common side or component to link the equations."}

Understanding Angle of Elevation vs. Angle of Depression

The two most critical concepts in this chapter are the Angle of Elevation and the Angle of Depression. Both angles are always measured with respect to a horizontal line extending from the observer's eye.

Imagine you are standing on the ground and looking up at a bird in the sky. Your line of sight to the bird is pointing upwards. The angle formed between your horizontal line of sight (looking straight ahead) and your upward line of sight to the bird is the Angle of Elevation. You elevate your gaze.

Now, imagine you are standing on the balcony of a tall building and looking down at a car on the road. Your line of sight to the car is pointing downwards. The angle formed between your horizontal line of sight (looking straight out) and your downward line of sight to the car is the Angle of Depression. Your gaze is depressed, or lowered.

A key property to remember is that if observer A on the ground looks up at observer B on a tower, the angle of elevation is 'θ'. If observer B looks down at observer A, the angle of depression is also 'θ'. This is because the two horizontal lines (one at A's eye level, one at B's) are parallel, and the line of sight acts as a transversal, making the angle of elevation and angle of depression alternate interior angles, which are always equal.

Step-by-Step Guide to Solving Height & Distance Problems

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Trigonometric Ratios Recap (SOH CAH TOA)

Quick Solved Examples

• {"problem":"The top of a tower is observed from a point on the ground, 20 m away from its foot. The angle of elevation is 60°. Find the height of the tower.","solution":"Let the height of the tower be 'h' and the distance be 'd'. Here, d = 20 m, angle θ = 60°. We have the opposite side (h) and adjacent side (d). So we use tan.\n tan 60° = h / d\n √3 = h / 20\n h = 20√3 m. \nSo, the height of the tower is 20√3 meters."}
• {"problem":"A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.","solution":"The ladder forms the hypotenuse (10 m), the wall height is the opposite side (8 m). We need to find the adjacent side (distance 'd'). We can use Pythagoras' theorem.\n (Hypotenuse)² = (Opposite)² + (Adjacent)²\n 10² = 8² + d²\n 100 = 64 + d²\n d² = 36\n d = 6 m. \nThe distance is 6 meters."}
• {"problem":"From the top of a 50 m high lighthouse, the angle of depression of a ship is 30°. Find the distance of the ship from the lighthouse.","solution":"Height of lighthouse (h) = 50 m. Angle of depression = 30°. The angle of elevation from the ship to the top of the lighthouse is also 30°. Let the distance be 'd'.\n tan 30° = h / d\n 1/√3 = 50 / d\n d = 50√3 m. \nThe ship is 50√3 meters away from the lighthouse."}

Exam Traps & Scoring Tips

Examiners often check three things in this chapter: your diagram, your choice of trigonometric ratio, and your calculation.

• Diagram is King: Always draw a diagram, even if it's not explicitly asked for. A correct, labeled diagram can fetch you marks even if your final calculation is wrong. A wrong diagram will almost certainly lead to a zero for the question.
• Angle of Depression Trap: A common mistake is marking the angle of depression inside the triangle. Remember, it's formed with the horizontal line outside the triangle. Use the alternate interior angle property to bring it inside the triangle as the angle of elevation.
• Rationalize the Denominator: If your answer is something like 50/√3, always rationalize it to (50√3)/3 unless specified otherwise.
• Use Given Values: If the question provides a value like √3 = 1.73, you must use it in your final step to get a decimal answer. Not doing so may lead to a loss of marks.

Practice Questions with Solutions

• If the angle of elevation of the sun is 45°, what is the relationship between the height of a vertical pole and the length of its shadow? They are equal. Since tan 45° = 1, and tan θ = height/shadow, height/shadow = 1, which means height = shadow.
• What is the angle formed by the line of sight with the horizontal when the object is viewed above the horizontal level? Angle of Elevation.
• If the angle of depression from a tower to an object is 60°, what is the angle of elevation of the top of the tower from that object? 60°, because the angle of elevation and angle of depression are alternate interior angles and are equal.
• A 10m ladder leaning against a wall makes an angle of 60° with the ground. How far is the foot of the ladder from the wall? 5m. Use cos 60° = Adjacent/Hypotenuse. cos 60° = distance/10. So, 1/2 = distance/10, which gives distance = 5m.

Frequently Asked Questions