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What is the formula for finding horizontal distance if the staff is held vertical and line of sight is inclined?

$D = k s cos^{2} \theta + c \sin \theta$

$D = k s \cos 2 \theta + c \sin \theta$

$D = k s cos^{2} \theta + c \cos \theta$

$D = k s \cos \theta ^{2} + c \cos \theta$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey

Option C

$D = k s cos^{2} \theta + c \cos \theta$

Quick Summary:

When the staff is held vertical and the line of sight is inclined at an angle $\theta$ to the horizontal, the horizontal distance $D$ between the instrument station and the staff station is given by $D = ks \cos^2\theta + c \cos\theta$ . Here, $k$ is the multiplying constant ( $F/i$ ), $c$ is the additive constant ( $F+d$ ), and $s$ is the staff intercept.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$D = ks \cos^2\theta + c \cos\theta$ — Horizontal distance for vertical staff and inclined line of sight

$V = \frac{1}{2} ks \sin 2\theta + c \sin\theta$ — Vertical component for vertical staff and inclined line of sight

$L = ks \cos\theta + c$ — Inclined distance along line of sight

⚙️ Working Principle

The line of sight is inclined at an angle $\theta$ , making the staff non-perpendicular to the line of sight. The effective staff intercept perpendicular to the line of sight becomes $s' = s \cos\theta$ . The inclined distance along the line of sight is $L = k s' + c = ks \cos\theta + c$ . Resolving $L$ horizontally gives the horizontal distance $D = L \cos\theta = ks \cos^2\theta + c \cos\theta$ .

📌 Key Points

▸
The multiplying constant $k = \frac{f}{i}$ is typically equal to 100 in modern tacheometers.
▸
The additive constant $c = f + d$ is made zero by using an anallactic telescope.
▸
When using an anallactic lens ( $c = 0$ ), the formula simplifies to $D = ks \cos^2\theta$ .

✅ Advantages

▸
Eliminates the need for direct chaining or taping over rough or hilly terrain.
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Allows rapid measurement of both horizontal distance and elevation differences.

❌ Disadvantages / Limitations

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Less precise than direct electronic distance measurement (EDM) or precise chaining.
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Accuracy drops significantly at high vertical angles ( $\theta > 15°$ ).

🛠️ Applications / Uses

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Topographic surveying in hilly or inaccessible terrain.
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Contour mapping and preparation of location plans.

📄 Additional Information

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Option A incorrect: Uses $\sin\theta$ for the additive component instead of $\cos\theta$ .
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Option B incorrect: Uses $\cos 2\theta$ instead of $\cos^2\theta$ and $\sin\theta$ instead of $\cos\theta$ .
▸
Option D incorrect: Places the exponent incorrectly on the angle $\theta$ rather than the cosine function.

📊 Diagram / Illustration

✅

C is correct — The horizontal distance formula for an inclined line of sight with a vertical staff is $D = ks \cos^2\theta + c \cos\theta$ .

Core Concepts Used

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Tacheometry Stadia System Anallactic Lens

💡 EXAM TIP

For anallactic telescopes ( $c = 0$ ), always use $D = ks \cos^2\theta$ and $V = \frac{1}{2}ks \sin 2\theta$ to save time during competitive exam calculations.

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