Trigonometry — Grade X

Grade X

TRIGONOMETRY || त्रिकोणमिति

Trigonometry त्रिकोणमिति

Math — Grade X · Chapter 7

📚 1 Topic

🎯 TQ 4 · TM 4

🔢 K:1 · U:1 · A:1 · HA:1

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Trigonometry

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Exercise — Model 1

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Key Formulas

Height & Distance — angles, ratios & standard values

Basic Trig Ratios

$$\sin\theta = \frac{\text{Opp}}{\text{Hyp}}, \quad \cos\theta = \frac{\text{Adj}}{\text{Hyp}}$$ $$\tan\theta = \frac{\text{Opp}}{\text{Adj}}$$

SOH CAH TOA — the foundation for all height & distance problems.

Height from Angle of Elevation

$$\tan\theta = \frac{\text{Height}}{\text{Distance}}$$ $$\text{Height} = \text{Distance} \times \tan\theta$$

Used when a man/observer looks UP at a pole, tower, kite or treetop.

Distance from Angle of Depression

$$\tan\theta = \frac{\text{Height difference}}{\text{Horizontal distance}}$$ $$d = \frac{h}{\tan\theta}$$

Used when an observer looks DOWN from a tower/pillar to a house or object below.

Effective Height (Pole & Man)

$$h = H_{\text{pole}} - H_{\text{man}}$$ $$\tan\theta = \frac{h}{d}$$

e.g. Pole = 20.5 m, Man = 1.5 m → effective height = 19 m. Always subtract observer's eye level.

Two-Angle (Pillar & House) Setup

$$H_{\text{pillar}} = d \cdot \tan\alpha$$ $$H_{\text{house}} = H_{\text{pillar}} - d \cdot \tan\beta$$

α = depression to base, β = depression to roof. Both from top of pillar. Distance d is the same.

Broken Tree Formula

$$\tan\theta = \frac{\text{Standing part}}{x}, \quad \cos\theta = \frac{x}{\text{Broken part}}$$ $$\text{Total height} = \text{Standing} + \text{Broken part}$$

x = distance from foot to where top touches ground. Use tan for standing height, cos for broken length.

Kite / String Problem

$$CH = AC \cdot \sin\theta$$ $$\text{Kite height} = CH + AB_{\text{boy}}$$

AC = string length, θ = angle with horizontal. Add boy's height to get total height above ground.

Circular Pond / Pole at Centre

$$r = \frac{C}{2\pi}, \quad \tan\theta = \frac{H_{\text{above water}}}{r}$$ $$H_{\text{above water}} = H_{\text{total}} - \text{depth}$$

Find radius from circumference. Subtract pond depth from pole height before applying tan.

Elevation = Depression (Alternate Angles)

$$\angle \text{elevation (from A to B)} = \angle \text{depression (from B to A)}$$

Alternate interior angles — the two horizontal lines are parallel, line of sight is the transversal.

Horizontal Distance Change

$$d_1 = \frac{h}{\tan\theta_1}, \quad d_2 = \frac{h}{\tan\theta_2}$$ $$\Delta d = d_1 - d_2$$

Height stays fixed. When angle increases, person is closer. Subtract distances to find how far to move.

Rope Length (Pole & Circular Ground)

$$\text{Rope} = \sqrt{r^2 + h^2}$$ $$\tan\theta = \frac{h}{r}$$

r = radius (from circumference). Use Pythagoras for rope length; tan for angle rope makes with ground.

Standard Angles — Table

θ	0°	30°	45°	60°	90°
sin	0	½	$\frac{1}{\sqrt2}$	$\frac{\sqrt3}{2}$	1
cos	1	$\frac{\sqrt3}{2}$	$\frac{1}{\sqrt2}$	½	0
tan	0	$\frac{1}{\sqrt3}$	1	$\sqrt3$	—

Memorise these — every height & distance problem uses one of these angles.

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Key Concepts

Understand the ideas behind every Height & Distance problem

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Angle of Elevation

The angle formed between the horizontal line and the line of sight when an observer looks upward at an object. Always measured from the horizontal upward. Examples: man looking up at a pole, kite, or tower top.

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Angle of Depression

The angle formed between the horizontal line and the line of sight when an observer looks downward at an object. Always measured from the horizontal downward. Examples: observer at tower top looking down at a house roof or base.

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Elevation = Depression (Why?)

The angle of elevation from A to B always equals the angle of depression from B to A. The two horizontal lines at A and B are parallel; the line of sight is a transversal — they create equal alternate interior angles.

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Always Subtract Observer's Height

When a person observes a pole or tower, the effective vertical difference = pole height − person's height. Draw a horizontal from the observer's eye level — the right triangle sits above that line, not from the ground up.

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Broken Tree Problems

The broken part = hypotenuse. Standing part = opposite side. Distance from foot to where top touches = adjacent side. Use tan θ to find standing height, cos θ for broken length. Total tree height = standing + broken part.

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Two-Object Problems (Pillar & House)

When angles of depression to both roof and base of a house are given from a pillar top, treat each angle separately. Pillar height = d × tan(base angle). House height = Pillar height − d × tan(roof angle). Horizontal distance d is shared by both triangles.

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Kite Problems

String = hypotenuse. Angle with horizontal = angle of elevation. Vertical rise = string length × sin θ. Always add the boy's height to get the kite's total height above ground. If the angle increases, the height increases — subtract the two heights to find the difference.

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Moving Closer / Further

Height stays fixed when the person moves. Find d₁ = h / tan θ₁ and d₂ = h / tan θ₂. Larger angle → person is closer. Subtract the two distances to find exactly how far forward or backward the person must walk.

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Circular Ground / Pond Problems

Find radius: r = C / 2π. Pole is at centre; observer at bank — so horizontal distance = radius. Subtract pond depth from total pole height to get height above water. Then tan θ = visible height / radius to find the angle.

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Problem-Solving Strategy

Step 1: Draw a clear diagram — label all heights, distances, and angles. Step 2: Identify the right triangle formed. Step 3: Choose the correct ratio (almost always tan). Step 4: Substitute and solve. Never skip the diagram — it instantly reveals which sides and angle to use.

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