Question

If θ₁ and θ₂ be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of dip θ is given by

• A. cot²θ = cot²θ₁ + cot²θ₂
• B. tan²θ = tan²θ₁ + tan²θ₂
• C. cot²θ = cot²θ₁ – cot²θ₂
• D. tan²θ = tan²θ₁ – tan²θ₂

Answer 1

The Correct Option is A

To determine the true angle of dip \( \theta \) in relation to the apparent angles of dip \( \theta_1 \) and \( \theta_2 \), we start by understanding that these are observed in two vertical planes that are perpendicular to each other. The relationship is derived from the vector components of the Earth's magnetic field.

Given:

• \( \theta_1 \): Apparent angle of dip in the first vertical plane. 
• \( \theta_2 \): Apparent angle of dip in the second vertical plane, perpendicular to the first.

The true angle of dip \( \theta \) can be expressed in terms of the apparent angles observed in each plane. The formula relating these angles is:

\(\cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2\)

Let's understand why this formula holds:

• The angle of dip is the angle that the Earth's magnetic field makes with the horizontal.
• When observations are made in two perpendicular planes, the mathematical relationship involving the cotangent arises due to the trigonometric identity and vector addition principles applied over components in these two planes.

Therefore, substituting the values of the apparent angles of dip into the cotangent form gives the true angle of dip. The correct option is:

 

cot²θ = cot²θ₁ + cot²θ₂

 

This conclusion rules out the other options which suggest subtraction or the use of tangent, as these do not hold under the conditions given.