Trigonometry for Officer Cadet: Maths Formulas, Tricks, and Practice Questions

Trigonometry is an important part of Mathematics that covers the Sine Law, area of a triangle using the Sine Law, height and distance concepts, and related problem-solving. It focuses on applying trigonometric principles to calculate unknown sides, angles, heights, and distances in practical problems.

For Officer Cadet Entrance Exam preparation, trigonometry questions help develop mathematical reasoning and the ability to apply trigonometric formulas accurately. This resource covers essential trigonometry formulas, solving methods, useful tricks, and practice questions related to the Sine Law, area of triangles, height and distance, and related applications. Explore the complete Officer Cadet Exam resources and Maths for Officer Cadet for additional practice and study materials.

What is Trigonometry?

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In the context of Officer Cadet exams, it is primarily used to solve problems involving:
👉 Finding the area of any triangle when height is not directly available
👉 Calculating inaccessible heights (mountains, towers, buildings)
👉 Determining distances across rivers, valleys, or between ships
The two pillars of Officer Cadet trigonometry are the Sine Law (for solving oblique triangles) and Height & Distance (for practical application problems).

Trigonometry for Officer Cadet: Syllabus

This Mathematics for Officer Cadet Entrance Exam Preparation resource covers the following main topics:
1. Area of Triangle by Using Sine Law
2. Concept of Height and Distance and Problem Solving

1. Area of Triangle by Using Sine Law

The Sine Law

Before calculating area, you must understand the Sine Law, which relates the sides of a triangle to the sines of its opposite angles.

Formula:
For a triangle with sides a, b, c and opposite angles A, B, C:$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$sinAa​=sinBb​=sinCc​

The Sine Law is used when you know either:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA)

Area Formula Using Sine

The most versatile area formula for triangles uses two sides and the included angle.

Formula:$$\text{Area}=\frac{1}{2}\times a\times b\times \sin C$$

Where a and b are two sides, and C is the angle between them .

Equivalently:
👉 Area = ½ bc sin A
👉 Area = ½ ac sin B

Example 1 (Basic Area Calculation):
Find the area of a triangle with sides 15.2 cm and 18.0 cm with an included angle of 62°.

Solution:$$\text{Area}=\frac{1}{2}\times 15.2\times 18.0\times \sin {62}^{\circ }$$$$=136.8\times 0.8829$$$$=120.8{\text{ cm}}^2$$

Example 2 (Using Sine Law First):
Given a = 10√3, b = 10, A = 120°, find the area of the triangle.

Solution:
First, use the Sine Law to find angle B:$$\frac{10\sqrt{3}}{\sin {120}^{\circ }}=\frac{10}{\sin B}$$$$\sin B=\frac{10\times \sin {120}^{\circ }}{10\sqrt{3}}=\frac{\sqrt{3}\mathrm{/}2}{\sqrt{3}}=\frac{1}{2}$$

So B = 30°, and C = 180° − (120° + 30°) = 30°.

Now apply the area formula:$$\text{Area}=\frac{1}{2}\times 10\sqrt{3}\times 10\times \sin {30}^{\circ }$$$$=50\sqrt{3}\times \frac{1}{2}=25\sqrt{3}$$

2. Concept of Height and Distance

Height and Distance problems use right-triangle trigonometry to find heights of tall objects or distances that cannot be measured directly.

Key Definitions

Angle of Elevation:
The angle formed between the horizontal line of sight and the line of sight to an object above the horizontal level .

Angle of Depression:
The angle formed between the horizontal line of sight and the line of sight to an object below the horizontal level .

Important Property:
By geometry, the angle of elevation from point A to point B equals the angle of depression from point B to point A (alternate interior angles).

Trigonometric Ratios for Right Triangles

For a right triangle with angle θ:$$\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$$$$\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}$$$$\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$$

Standard Values

Example 1 (Basic Height Calculation):
From a point 34 m away from the base of a building, the angle of elevation to the top is 65°. Find the height of the building.

Solution:$$\tan {65}^{\circ }=\frac{h}{34}$$$$h=34\times \tan {65}^{\circ }$$$$h=34\times 2.1445=72.9\text{ m}$$

Example 2 (Two Angles Problem):
The angle of elevation of a tree changes from 60° to 30° after walking 20 m away from it. Find the height of the tree.

Solution:
Let the height be h, and let the initial distance from the base be x.

From the first position:$$\tan {60}^{\circ }=\frac{h}{x}\Rightarrow \sqrt{3}=\frac{h}{x}\Rightarrow x=\frac{h}{\sqrt{3}}$$

From the second position:$$\tan {30}^{\circ }=\frac{h}{x+20}\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+20}$$$$x+20=h\sqrt{3}$$

Substituting x = h/√3:$$\frac{h}{\sqrt{3}}+20=h\sqrt{3}$$$$20=h\sqrt{3}-\frac{h}{\sqrt{3}}=h\left(\frac{3-1}{\sqrt{3}}\right)=\frac{2h}{\sqrt{3}}$$$$h=10\sqrt{3}\approx 17.32\text{ m}$$

Example 3 (Angle of Depression):
From the top of a cliff 92 m high, the angle of depression of a buoy is 20°. Find the distance of the buoy from the foot of the cliff.

Solution:
The angle of elevation from the buoy to the cliff top equals the angle of depression (20°).$$\tan {20}^{\circ }=\frac{92}{d}$$$$d=\frac{92}{\tan {20}^{\circ }}=\frac{92}{0.3640}=252.7\text{ m}$$

Effective Tricks to Solve Trigonometry Questions

Trick 1: Draw a Clear Diagram First
Before writing any equation, sketch the scenario. Mark the right angles, label known sides and angles, and identify what you need to find. A good diagram prevents most mistakes.

Trick 2: Identify Whether You Need Sine Law or Right-Triangle Ratios
👉 If the triangle has a right angle, use basic ratios (sin, cos, tan).
👉 If the triangle has no right angle, use the Sine Law or area formula.

Trick 3: For Two Angles with a Known Distance Between Them
When the angle of elevation changes from θ₁ to θ₂ after moving a distance d, use:$$h=\frac{d}{\cot {\theta }_2-\cot {\theta }_1}$$​

Trick 4: Remember the “Elevation = Depression” Property
When you see an angle of depression, immediately transfer it as an angle of elevation at the object’s location. This often makes the problem simpler .

Trick 5: Use the Complement Rule for “At Least” Problems
For problems involving heights and distances, if the angle is very small, the distance is very large. Use approximations only when necessary.

Trick 6: Check Units Consistency
Ensure all distances are in the same unit (meters, feet, km) before performing calculations.

Important Tips to Solve Trigonometry Questions

👉 Memorise standard angles: Know sin, cos, and tan for 0°, 30°, 45°, 60°, and 90° by heart.

👉 Draw to scale when possible: A roughly scaled diagram helps you verify whether your answer makes sense.

👉 Watch for “from the top” vs “from the foot”: The reference point determines whether you use elevation or depression.

👉 Use exact values until the final step: Keep √3 as √3 rather than approximating early to maintain accuracy.

👉 Practise problems with two observers: These are the most common exam questions and test your ability to set up simultaneous equations.

👉 Time yourself: Trigonometry problems can be solved quickly with practice, so build speed through repetition.

Practice Questions

Question 1:
Find the area of a triangle with sides 8 cm and 12 cm and an included angle of 45°.

Question 2:
From a point 50 m from the foot of a tower, the angle of elevation of the top is 30°. Find the height of the tower.

Question 3:
The angle of elevation of the top of a hill from a point is 45°. After walking 100 m away, the angle becomes 30°. Find the height of the hill.

Question 4:
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the wall.

Question 5:
From the top of a lighthouse 80 m high, the angles of depression of two ships on opposite sides are 30° and 45°. Find the distance between the ships.

Question 6:
In triangle ABC, a = 15, b = 20, and C = 60°. Find the area of the triangle.

Question 7:
The angle of elevation of a cloud from a point 60 m above a lake is 30°, and the angle of depression of its reflection is 60°. Find the height of the cloud above the lake.

Conclusion

Trigonometry is a scoring and practical topic in the mathematics syllabus for Officer Cadet Entrance Exam preparation. The two core areas—Area of Triangle using Sine Law and Height & Distance—appear consistently in exams and reward students who can visualise problems and apply the right formulas efficiently.

Focus on mastering the Sine Law and the standard trigonometric ratios, practise drawing accurate diagrams for every problem, and learn to recognise common patterns like two-angle problems and angle of depression scenarios. With regular practice, trigonometry can become a quick-scoring area in the Mathematics section.

This resource is part of the Officer Cadet Entrance Exam Preparation series, focusing on Maths for Officer Cadet Entrance Exam. For structured lessons, comprehensive practice question banks, and mock tests, explore the Trigonometry section of the Maths for Officer Cadet Entrance Exam course on the Pratima Academy LMS.

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