Trigonometry & Inverse Trigonometry
Heights And Distances
nta_abhyas_2025
Grade 11
Question:
$AB$ is a vertical tower. The point $A$ is on the ground and $C$ is the middle point of $AB$. The part $CB$ subtends an angle $\alpha$ at a point $P$ on the ground. If $AP = n \cdot AC$, then the correct relation is
n = (n^2 + 1) \tan \alpha
n = (2n^2 - 1) \tan \alpha
n^2 = (2n^2 + 1) \tan \alpha
n = (2n^2 + 1) \tan \alpha
Step-by-Step Solution
Key Concept: The tangent difference formula decomposes composite angles into simpler trigonometric expressions
Using the tangent difference formula, $\tan \alpha = \tan(\beta - \theta)$ where $\tan \beta = \frac{AB}{AP}$ and $\tan \theta = \frac{AC}{AP}$. Applying the formula: $\tan \alpha = \frac{\frac{AB}{AP} - \frac{AC}{AP}}{1 + \frac{AB \cdot AC}{AP^2}} = \frac{AB - AC}{AP + \frac{AB \cdot AC}{AP}}$. Since $AP = n(2AC) = 2n \cdot AC$, we get $n = (2n^2 + 1)\tan \alpha$.
Correct Answer: 2