Chapter 9: Applications of Trigonometry - Board Exam Notes

Exam Weightage & Blueprint

Total: ~4-6 Marks

This chapter is famous for Case Studies and Long Answer Questions. It's high scoring if the diagram is correct.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	Low	Simple height finding ($\tan \theta$)
Short Answer	3	Medium	Single Triangle Problems (Broken Tree, etc.)
Long Answer / Case Study	4-5	Very High	Double Triangle Problems (Two ships, Building & Tower)

⏰ Last 24-Hour Checklist

Angle of Elevation: Looking UP from horizontal.
Angle of Depression: Looking DOWN from horizontal.
Key Relation: Angle of Depression = Angle of Elevation (Alternate Angles).

Values: Memorize $\tan 30^\circ, \tan 45^\circ, \tan 60^\circ$.
Diagram Rule: Always mark the right angle and the reference angle.
Root Values: $\sqrt{3} = 1.732$ (Use only if asked).

Core Theory & Definitions ★★★★★

Line of Sight: The line drawn from the eye of an observer to the point in the object viewed by the observer.

Angles Explained

Angle of Elevation

The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.

Angle of Depression

The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.

Important: In Angle of Depression problems, always draw a horizontal line from the observer's eye level first. Then mark the angle downwards.

Essential Formulas & Values 🔥🔥🔥

Most questions are solved using the tangent ratio:

$$ \tan \theta = \frac{\text{Perpendicular (Height)}}{\text{Base (Distance)}} $$

Values to Remember

30°
$\tan 30^\circ = \frac{1}{\sqrt{3}}$

45°
$\tan 45^\circ = 1$

60°
$\tan 60^\circ = \sqrt{3}$

Solved Examples (Board Marking Scheme)

Q1. A tower stands vertically on the ground. From a point 15 m away from the foot, the angle of elevation is $60^\circ$. Find the height. (2 Marks)

Step 1: Diagram & Given 0.5 Mark

Let AB be the tower ($h$). Point C is 15 m away. $\angle ACB = 60^\circ$.

Step 2: Apply Ratio 1 Mark

In $\triangle ABC$, $\tan 60^\circ = \frac{AB}{BC} = \frac{h}{15}$.

Step 3: Solve 0.5 Mark

$\sqrt{3} = \frac{h}{15} \implies h = 15\sqrt{3}$ m.

Q2. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower. (5 Marks)

Step 1: Setup & Diagram 1 Mark

Let AB be the building (7m) and CD be the tower ($H$). Draw horizontal line AE.

$\angle CAE = 60^\circ$ (Elevation), $\angle EAD = 45^\circ$ (Depression).

Step 2: Triangle 1 (Lower) 1.5 Marks

In $\triangle ABD$, $\tan 45^\circ = \frac{AB}{BD} = \frac{7}{BD}$.

$1 = \frac{7}{BD} \implies BD = 7$ m.

Also, $AE = BD = 7$ m.

Step 3: Triangle 2 (Upper) 1.5 Marks

In $\triangle AEC$, $\tan 60^\circ = \frac{CE}{AE}$.

$\sqrt{3} = \frac{CE}{7} \implies CE = 7\sqrt{3}$ m.

Step 4: Total Height 1 Mark

Height of tower = $CE + ED = 7\sqrt{3} + 7 = 7(\sqrt{3} + 1)$ m.

Previous Year Questions (PYQs)

2023 (Case Study): A kite is flying at a height of 60m. The string makes an angle of $60^\circ$ with the ground. Find string length.
Ans: Use $\sin 60^\circ = \frac{P}{H} \implies \frac{\sqrt{3}}{2} = \frac{60}{L} \implies L = \frac{120}{\sqrt{3}} = 40\sqrt{3}$ m.

2020 (3 Marks): The shadow of a tower is $40$m longer when the Sun's altitude is $30^\circ$ than when it is $60^\circ$. Find the height.
Ans: $h(\cot 30^\circ - \cot 60^\circ) = 40 \implies h(\sqrt{3} - \frac{1}{\sqrt{3}}) = 40 \implies h = 20\sqrt{3}$ m.

2019 (4 Marks): Two poles of equal heights stand on either side of an 80m wide road. Angles of elevation are $60^\circ$ and $30^\circ$. Find height and distances.
Ans: $h = 20\sqrt{3}$ m. Distances are 20m and 60m.

Exam Strategy & Mistake Bank

Mistake Bank 🚨

Wrong Ratio: Using $\tan$ when hypotenuse (ladder/kite string) is involved. Use $\sin$ or $\cos$ instead.

Angle Place: Marking the angle of depression with the vertical instead of the horizontal.

Calculation: $\sqrt{3} \approx 1.732$. Use this value only if specified in the question.

Scoring Tips 🏆

Diagram is Mandatory: Even if you solve it correctly, no marks for diagram = marks deducted.

Label Vertices: Always label the triangle vertices (A, B, C) and state "Let AB be the tower...".

Double Triangles: In 5-mark questions, you usually get two equations with 'h' and 'x'. Isolate 'x' in one and substitute in the other.

Self-Assessment Mock Test (10 Marks)

Q1 (1M): The angle of elevation of the sun when the shadow of a pole is equal to its height is _______.

Q2 (2M): A ladder 15m long makes an angle of $60^\circ$ with the wall. Find the height of the point where the ladder touches the wall.

Q3 (3M): A tree breaks due to a storm and the broken part bends so that the top touches the ground making an angle $30^\circ$. Distance is 8m. Find total height.

Q4 (4M): From the top of a 75m high lighthouse, the angles of depression of two ships are $30^\circ$ and $45^\circ$. If one ship is exactly behind the other, find the distance between them.