Theodolite Angle Error Determination

The objective is to find the linear offset required to correct a $90^\circ$ angle due to a measurement error detected using a theodolite.

Angular Error Calculation

The difference between the measured angle and the true angle gives the angular error.

• Measured Angle: $90^\circ 00' 25''$
• True Angle (Target): $90^\circ 00' 00''$
• Angular Error ($\Delta\theta$): $90^\circ 00' 25'' - 90^\circ 00' 00'' = 0^\circ 00' 25''$

Convert Angular Error to Radians

For calculating linear offsets, the angular error must be expressed in radians.

• Error in seconds: $25''$
• Conversion factor: $1'' = \frac{\pi}{180 \times 3600}$ radians
• Angular Error in Radians ($\delta\theta$): $25 \times \frac{\pi}{180 \times 3600} = \frac{25\pi}{648000}$ radians

Calculate Linear Offset

The linear offset ($x$) is approximately the product of the distance ($D$) and the angular error in radians ($\delta\theta$), using the small angle approximation ($x \approx D \times \delta\theta$).

• Distance ($D$): $300$ m
• Linear Offset ($x$): $300 \text{ m} \times \frac{25\pi}{648000}$
• $x \approx \frac{7500\pi}{648000} = \frac{\pi}{86.4}$ m
• Using $\pi \approx 3.14159265$:
• $x \approx \frac{3.14159265}{86.4} \approx 0.0363553$ m

Final Offset Value

Rounding the calculated offset to three decimal places provides the final answer.

• Rounded Offset: $0.036$ m

The calculated value $0.036$ m falls within the expected range of $0.035$ m to $0.037$ m.