Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

From the top of a tower \(100 \mathrm{~m}\) height, the angles of depression of two objects \(200 \mathrm{~m}\) apart on the horizontal plane and in a line passing through the foot of the tower and on the same side of the tower are \(45^{\circ}-A\) and \(45^{\circ}+A\). The angle \(\mathrm{A}\) is equal to (A) \(15^{\circ}\) (B) \(35^{\circ}\) (C) \(22 \frac{1}{2}^{\circ}\). (D) \(45^{\circ}\)

Step-by-Step Solution

Answer

The angle \(A\) is \(22.5^{\circ}\).

Step 1: Understanding the Setup

We have a tower that is 100 meters tall. From the top of this tower, two objects are observed. These objects form angles of depression of \(45^{\circ}-A\) and \(45^{\circ}+A\), and they are 200 meters apart on the ground. We need to find the value of \(A\).

Step 2: Relating Tower Height and Distance Using Trigonometry

For an angle of depression \(\theta\), the horizontal distance \(d\) between the object and the base of the tower can be calculated using the formula: \(d = h \cdot \cot(\theta)\). Here, \(h\) is the height of the tower, which is 100 meters.

Step 3: Calculate Distances for Both Angles

For the angle of depression \(45^{\circ} - A\), the distance to the first object is: \[d_1 = 100 \cdot \cot(45^{\circ} - A)\] For \(45^{\circ} + A\), the distance to the second object is: \[d_2 = 100 \cdot \cot(45^{\circ} + A)\]

Step 4: Relate Distances to Known Separation

It is given that the distance between the two objects is 200 meters. Therefore, we set up the equation: \[d_2 - d_1 = 200\] Which simplifies to:\[100 \cdot \cot(45^{\circ} + A) - 100 \cdot \cot(45^{\circ} - A) = 200\] Simplifying: \[\cot(45^{\circ} + A) - \cot(45^{\circ} - A) = 2\]

Step 5: Use Trigonometric Identity for Difference of Cotangents

The identity for the difference of cotangents is:\[\cot(x) - \cot(y) = \frac{\sin(y-x)}{\sin(x)\sin(y)} \cdot (\cos(x+y))\] Applying it here: \[\frac{2 \sin(2A)}{(1 + \tan^2(45^{\circ} - A))(1 + \tan^2(45^{\circ} + A))} = 2\] Since \(1 + \tan^2(x) = \sec^2(x)\): \[\tan(90^{\circ}-2A) = 2\] \[2A = 45^{\circ}\]

Step 6: Solve for A

From \(2A = 45^{\circ}\), we get \[A = \frac{45^{\circ}}{2} = 22.5^{\circ}\]

Key Concepts

Angles of Depression

The angle of depression is a practical concept used in trigonometry, especially when observing objects from a high point. It refers to the angle formed between the horizontal plane and the line of sight when looking down at an object. If you imagine standing on top of a tower and looking down at an object on the ground, the angle your line of sight makes with the horizontal plane is the angle of depression. This angle helps us understand the relationship between the height of the viewpoint and the distance of the object on the ground.

• It is equal to the angle of elevation when looking at the object from the ground, thanks to the alternate interior angles theorem in parallel lines.
• This angle is crucial in calculating distances and heights using trigonometry.

In our problem, angles of depression are given as \(45^{\circ} - A\) and \(45^{\circ} + A\). These angles help us to find the horizontal distances to the objects when combined with trigonometric functions.

Cotangent Function

The cotangent function, often abbreviated as "cot," is one of the basic trigonometric functions used to understand angles and their relationships in a triangle. It is the reciprocal of the tangent function. If you have an angle \(\theta\) in a right triangle:\[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent Side}}{\text{Opposite Side}}\]The cotangent function is especially useful when dealing with angles of depression or elevation, because it relates the height (or depth) of an observation point to the distance to the object viewed. This function allows us to compute such distances easily, provided the angle of observation is known.

• Cotangent is the ratio of the length of the adjacent side to the length of the opposite side in a right-angled triangle.
• In our exercise, the horizontal distances from the tower to both objects can be calculated using cotangent, given the angles of depression.

By applying this function, we calculated the distances from the top of the tower to the objects based on their angles of depression.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are fundamental for simplifying expressions and solving trigonometric equations. These identities come in handy in complex calculations, like the problem we are dealing with, where multiple angles and their trigonometric functions are involved.

Difference of Cotangent Identity

We used the identity for the difference of cotangents, which helps in expressing and simplifying the difference between two cotangent values:\[\cot(x) - \cot(y) = \frac{\sin(y-x)}{\sin(x)\sin(y)} \cdot (\cos(x+y))\]This identity simplifies our expressions to make calculation easier, especially when working with multiple angles like \(45^{\circ} + A\) and \(45^{\circ} - A\).

• These identities enable us to solve for unknowns, simplifying the algebra involved in trigonometric expressions.
• They bridge the gap between complex expressions and simpler terms that are easier to manage.

In this exercise, such identities allowed us to transform the equation into a more manageable form, ultimately leading us to solve for the angle \(A\).