Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The angle of elevation of a cloud from a point \(h\) metres above the surface of a lake is \(\theta\) and the angles of depression of its reflection is \(\phi\). Then the height of the cloud is (A) \(\frac{h \sin (\phi+\theta)}{\sin (\phi-\theta)}\) (B) \(\frac{h}{\sin (\phi-\theta)}\) (C) \(h \tan (\phi-\theta)\) (D) \(\frac{h \sin (\phi-\theta)}{\sin (\phi+\theta)}\)

Step-by-Step Solution

Answer

The height of the cloud is \(\frac{h \sin(\phi + \theta)}{\sin(\phi - \theta)}\).

Step 1: Understand the Problem Setup

The problem involves a cloud above a lake and its reflection. The angle of elevation from a point above the lake to the cloud is \(\theta\), and the angle of depression to the cloud's reflection is \(\phi\). We need to find the actual height of the cloud from the surface of the lake.

Step 2: Visualize the Scenario

Imagine a point, say \(P\), which is \(h\) meters above the lake surface. From this point, the angle upwards to the cloud is \(\theta\). If the cloud is directly above the lake, the reflection would appear directly below it. From the same point \(P\), the angle of depression to the reflection in the water is \(\phi\).

Step 3: Relate Angles to Heights

Let the cloud be at height \(H\) above the water. From point \(P\), where the elevation angle to the cloud is \(\theta\), we can express the height \(H - h\) as: \[(H - h) = \tan(\theta) \times x\] where \(x\) is the horizontal distance from point \(P\) over which the cloud is seen.

Step 4: Setup Equations for Reflection

Considering the reflection, the depth of the cloud's reflection below the surface can also be expressed in terms of \(x\). The height from point \(P\) to this reflected point can be expressed using the depression angle \(\phi\): \[(H + h) = \tan(\phi) \times x\] since the horizontal distance \(x\) remains constant.

Step 5: Solve Simultaneous Equations

We have two equations: \((H - h) = \tan(\theta) \times x\) and \((H + h) = \tan(\phi) \times x\). Solving these together will eliminate \(x\) and give us the height \(H\). Divide the two equations:\[\frac{H - h}{H + h} = \frac{\tan(\theta)}{\tan(\phi)}\]Cross-multiply to find \(H\):\[H \tan(\phi) - h \tan(\phi) = H \tan(\theta) + h \tan(\theta)\]

Step 6: Simplify and Solve for Cloud Height

Reorder terms to isolate \(H\):\[H \tan(\phi) - H \tan(\theta) = h \tan(\phi) + h \tan(\theta)\]Factor \(H\) on the left, and \(h\) on the right:\[H(\tan(\phi) - \tan(\theta)) = h(\tan(\phi) + \tan(\theta))\]Solving for \(H\):\[H = \frac{h(\tan(\phi) + \tan(\theta))}{\tan(\phi) - \tan(\theta)}\]Examine the problem based on available options, the height calculated is consistent with option A when further examined with trigonometric sums: \[H = \frac{h \sin(\phi + \theta)}{\sin(\phi - \theta)}\]. This can be correlated using auxiliary angle transformations.

Key Concepts

Angle of Elevation

The angle of elevation is a fundamental concept in trigonometry. Let's imagine you are standing at a point above the ground. When you look up to an object located above your sightline, the angle formed between your sightline and the horizontal is known as the angle of elevation. In our problem, this angle is noted as \(\theta\). This angle helps us relate the horizontal distance to the height of an object, which in this case, is the cloud.
For instance, if you are at point \(P\) above a lake and gaze upwards at the cloud, creating an angle of elevation \(\theta\) with your horizontal line of sight, this can be represented using the tangent function:

• The vertical rise \((H - h)\) relates to the angle of elevation through the equation \((H - h) = \tan(\theta) \times x\).
• Here, \(x\) denotes the horizontal distance from point \(P\) to the point directly below the cloud.

Understanding the angle of elevation means grasping these relationships between angle, vertical height change, and horizontal projection.

Angle of Depression

The angle of depression is closely tied to the concept of the angle of elevation. Unlike the angle of elevation, it involves looking downward. From a higher point, such as point \(P\), looking down towards the reflection of a cloud on a lake, the angle that one's line of sight makes with the horizontal is termed the angle of depression.
In the exercise, it is noted as \(\phi\). This concept is vital for calculating how far below the surface a reflection appears, considering its actual position above the surface.

• If you are standing at point \(P\) and observe the cloud's reflection downward with an angle of depression \(\phi\), the downward angle helps express the original height plus the distance above height, creating another useful equation in our calculations.
• The relation is established by the equation \((H + h) = \tan(\phi) \times x\), where the height \(H\) plus \(h\) is the rise when looking at the reflection, and \(x\) remains the horizontal distance.

Angles of depression and elevation function together by linking the observer's perspective from point \(P\) to the positions of both the actual object and its reflection on the surface.

Reflection

Reflection in trigonometry refers to the mirror image formed in surfaces like lakes or mirrors. It is a captivating way to understand positions and heights. In our problem, the cloud's reflection in the lake creates another geometric consideration. From point \(P\), the reflection seems below the surface, though it's merely an inverted position of what is seen above.
A significant factor in this problem is understanding the constant horizontal distance, \(x\), which remains unaffected whether it's measuring the path to the cloud above or its reflection below.

• The equations \((H - h) = \tan(\theta) \times x\) and \((H + h) = \tan(\phi) \times x\) mirror each other with \(\theta\) and \(\phi\), managing the different perspectives to and from the actual position and reflection.
• Reflection, by using these angles, allows us to perceive and calculate distances indirectly, offering a unique measurement utility that simplifies calculating heights.

Reflect on these reflections, pun intended, to unveil the hidden information about heights and distances utilizing geometric insight.

Height Calculation

Calculating height using trigonometric principles is a cerebral exercise incorporating angles and reflections. It unveils exact positions using indirect observations. Our goal is to determine the cloud's actual height above the lake surface, where point \(P\) sits at height \(h\). We have equations representing heights using angles \(\theta\) and \(\phi\):

• By observing how these angles help relate to the horizontal distance \(x\), we set up two key equations: \((H - h) = \tan(\theta) \times x\) and \((H + h) = \tan(\phi) \times x\).
• Solving both equations simultaneously, we find our height calculation focuses on eliminating variable \(x\) and isolating \(H\), leading to the discovery of \(H = \frac{h(\tan(\phi) + \tan(\theta))}{\tan(\phi) - \tan(\theta)}\).

This expression is not only derived but further simplified using trigonometric identities to reach the solution:

• Via interpretations and sum identities, it can be concluded as \(H = \frac{h \sin(\phi + \theta)}{\sin(\phi - \theta)}\).
• Such transformations exhibit how angles governed by trigonometric functions seamlessly integrate to unveil heights with accuracy.

In conclusion, height calculation converges intuitive geometry with mathematics to unravel complex distances and heights.