Applications of Trigonometry

📚 Class 10 Mathematics – Applications of Trigonometry

Quick Revision Sheet (English – with simple Hindi notes)

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1️⃣ What is Trigonometry?

• Trigonometry (त्रिकोणमिति) – the study of relationships between the angles and sides of a right‑angled triangle.
• In this chapter we use those relationships to find heights and distances of objects that are not directly measurable.

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2️⃣ Basic Trigonometric Ratios

Ratio	Symbol	Definition (right‑angled triangle)	Hindi note
Sine	sin θ	opposite / hypotenuse = P / H	साइन = विपरीत / कर्ण
Cosine	cos θ	adjacent / hypotenuse = B / H	कोसाइन = समीप / कर्ण
Tangent	tan θ	opposite / adjacent = P / B	टैन्जेंट = विपरीत / समीप
Cotangent	cot θ	adjacent / opposite = B / P	कोटैन्जेंट = समीप / विपरीत
Secant	sec θ	hypotenuse / adjacent = H / B	सेकेंट = कर्ण / समीप
Cosecant	cosec θ	hypotenuse / opposite = H / P	कोसेकेंट = कर्ण / विपरीत

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3️⃣ Fundamental Trigonometric Identities

[ \begin{aligned} \sin^{2}\theta + \cos^{2}\theta &= 1 \ 1 + \tan^{2}\theta &= \sec^{2}\theta \ 1 + \cot^{2}\theta &= \cosec^{2}\theta \end{aligned} ]

These are useful for converting one ratio into another when solving problems.

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4️⃣ Angle of Elevation & Angle of Depression

Term	Definition	Diagram idea
Angle of Elevation	The angle measured upwards from the horizontal line of sight to an object above the horizontal.	Observer → horizontal → upward line to object
Angle of Depression	The angle measured downwards from the horizontal line of sight to an object below the horizontal.	Observer → horizontal → downward line to object

In calculations the horizontal line is taken as the reference; the same trigonometric ratios apply.

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5️⃣ Core Height‑&‑Distance Formulas

Situation	Formula	How it is derived
Height of an object (when distance from foot & angle of elevation are known)	[
h = d \tan\theta
]	tan θ = opposite/adjacent = h/d
Distance from the object (when height & angle of elevation are known)	[
d = \frac{h}{\tan\theta}
]	Rearranged from the previous line
Using cotangent (often convenient)	[
d = h \cot\theta
]	Since cot θ = 1/tan θ
Height when angle of depression is given (observer at height H)	[
h = H - d \tan\theta
]	Subtract the vertical drop from observer’s height
Height of a tower from two points (angles θ₁, θ₂ at distances d₁, d₂)	[
h = \frac{d_1 d_2 (\tan\theta_1 \tan\theta_2)}{d_2 \tan\theta_1 - d_1 \tan\theta_2}
]	From two right‑triangles sharing the same vertical side
Distance between two points on the same level (angles θ₁, θ₂ from a common point)	[
\text{Distance} = \frac{h(\tan\theta_1 + \tan\theta_2)}{\tan\theta_1 \tan\theta_2}
]	Useful for “two‑point” problems

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6️⃣ Step‑by‑Step Method to Solve Height‑&‑Distance Problems

1. Draw a clear diagram – label all known sides, angles, and the unknown you need.
2. Identify the right‑angled triangle(s) formed by the line of sight, the horizontal, and the vertical.
3. Choose the appropriate trigonometric ratio (sin, cos, tan, cot) based on the known side(s).
4. Write the equation using the chosen ratio.
5. Solve for the unknown – rearrange algebraically; use identities if needed.
6. Check units (usually metres) and whether the answer makes sense (positive, reasonable magnitude).

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7️⃣ Quick Reference Table

Symbol	Meaning	Formula (when useful)
θ	Angle of elevation / depression	–
h	Height of the object (or vertical difference)	h = d tanθ
d	Horizontal distance from observer to foot of object	d = h cotθ
H	Height of observer (or a known point)	–
P	Opposite side (vertical)	–
B	Adjacent side (horizontal)	–
H	Hypotenuse (line of sight)	–

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8️⃣ Commonly Asked Types of Problems

Problem Type	Typical Given Data	What to Find	Key Formula
Single point – elevation	Height of tree h or distance d, angle θ	Missing height or distance	h = d tanθ or d = h cotθ
Two points – same object	Two distances d₁, d₂ from foot, angles θ₁, θ₂	Height of object	h = (d₁ d₂ (tanθ₁ tanθ₂))/(d₂ tanθ₁ - d₁ tanθ₂)
Two points – same height	Height of object h, angles θ₁, θ₂ from two positions	Distance between the two positions	Δx = h (cotθ₁ - cotθ₂)
Angle of depression from a tower	Observer height H, angle θ	Horizontal distance to foot	d = H cotθ
Finding angle	Height h and distance d known	Angle of elevation	θ = tan⁻¹ (h/d)

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9️⃣ Tips & Tricks

• Always start with a diagram – a picture reduces mistakes.
• Use cotθ when the given angle is in the denominator (makes algebra cleaner).
• Check the quadrant – angles of elevation & depression are always acute (0° – 90°).
• Units must be consistent (all in metres or all in centimetres).
• Round off only at the final step to avoid cumulative errors.

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📌 Summary of Must‑Know Formulas

[ \begin{aligned} \sin\theta &= \frac{P}{H} \ \cos\theta &= \frac{B}{H} \ \tan\theta &= \frac{P}{B} \ \cot\theta &= \frac{B}{P} \ \sec\theta &= \frac{H}{B} \ \cosec\theta &= \frac{H}{P} \ \sin^{2}\theta + \cos^{2}\theta &= 1 \ 1 + \tan^{2}\theta &= \sec^{2}\theta \ 1 + \cot^{2}\theta &= \cosec^{2}\theta \ h &= d \tan\theta \ d &= \frac{h}{\tan\theta}=h\cot\theta \ \theta &= \tan^{-1}!\left(\frac{h}{d}\right) \end{aligned} ]

Keep this sheet handy while solving any “height and distance” problem – it contains every definition, law, and formula you’ll need for the Applications of Trigonometry chapter. Happy revising! 🚀