Problem 31 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\sin \theta=k\) for exactly one value of \(\theta, \theta \in\left[0, \frac{7 \pi}{3}\right]\), then the value of \(k\) is (A) 1 (B) \(-1\) (C) \(1 / \sqrt{2}\) (D) 0

Step-by-Step Solution

Answer

The value of \(k\) is 0.

Step 1: Understand the range of \( \theta \)

The problem states that \( \theta \) is in the interval \([0, \frac{7\pi}{3}])\). This interval goes beyond one complete revolution (\(2\pi\)), hence it's larger than just one cycle of sine function.

Step 2: Analyze one cycle of the sine function

The sine function \( \sin \theta \) has a range of \([-1, 1]\) and a period of \(2\pi\). Within a first complete cycle, i.e., \([0, 2\pi)]\), it takes all values from \(-1\) to \(1\) exactly twice for \( \theta \) within \( 0 \leq \theta \leq \pi \) and \( \pi < \theta < 2\pi \).

Step 3: Extend analysis to the interval \(\mathbf{[0, \frac{7\pi}{3}]}\)

The interval \([0, \frac{7\pi}{3}]\) equates to one full cycle \([0, 2\pi]\) plus an additional \(\frac{\pi}{3}\). The second cycle \([2\pi, \frac{7\pi}{3}]\) will cover sine values one more time starting from \(\sin(2\pi) = 0\) to \(\sin(\frac{7\pi}{3}) = \sin(\frac{\pi}{3})\).

Step 4: Determine when sine is unique in this interval

We need to find \(k\) such that it appears exactly once in \([0, \frac{7\pi}{3})\). In the original cycle \([0, 2\pi]\), \(\sin \theta\) hits all values in \([-1, 1]\) twice except at \(\sin(0) = 0\) and \(\sin(\pi) = 0\) which it hits once before repeating values as the second cycle starts at \([2\pi, \frac{7\pi}{3}]\).

Step 5: Conclusion

Given that \(\sin \theta = 0\) only exactly once from \([0, \pi) \) and exactly once again from \([2\pi, \frac{7\pi}{3})\), and we are not counting \(2\pi\) which is outside our domain, \(k = 0\) corresponds to exactly one value within \([0, \frac{7\pi}{3})\) interval.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry that expresses the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is written as \( \sin \theta \). The sine function has several key properties that are important to understand:

• Range: The range of the sine function is \([-1, 1]\). This means the possible values for \( \sin \theta \) are always between \(-1\) and \(1\).
• Domain: While the sine can take any real number as an input, in trigonometry we often deal with angles measured in radians, such as \( \theta \in [0, 2\pi] \).
• Zero points: The sine function equals zero at several points, including \(0, \pi,\) and \(2\pi\).

The sine function is a building block of periodic phenomena, which occurs naturally in many contexts, such as waves and oscillations. Understanding its values and behavior over an interval helps to decipher more complex trigonometric problems.
By knowing the sine function hits some values twice within a typical period can lead us to deduce unique scenarios such as in the given exercise.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. The sine function is a classic example of a periodic function.
Here's why:

• Period: The period of \( \sin \theta \) is \(2\pi\). This means every \(2\pi\) units, the function repeats itself. A sine wave will go through a full cycle from \( 0 \) to \(2\pi\), reach the same values again as it begins the next cycle.
• Cycle Repetition: This repetition ensures that the value pattern seen between \(0\) and \(2\pi\) will recur from \(2\pi\) to \(4\pi\), and so forth.

Every periodic function, including sine, helps in predicting and analyzing repeating patterns in different fields such as physics, engineering, and even daily life phenomena.
This exercise demonstrates the importance of knowing when and how often certain values are taken, notably when assessing the uniqueness of a value within a given range or interval.

Mathematical Intervals

An interval is a set of numbers lying between two numbers, often representing a part of the domain for a function. In trigonometry, intervals help define the scope within which functions are explored.
For example, the interval \([0, \frac{7\pi}{3}]\) includes all angles \( \theta \) starting from \(0\) to \(\frac{7\pi}{3}\), which is slightly more than two full cycles of the sine function.

• Types of intervals:
  • Closed intervals, denoted as \([a, b]\), include both endpoint values.
  • Open intervals, \((a, b)\), exclude the endpoint values.
  • Half-open intervals, such as \([a, b)\), include one endpoint but not the other.
• Usage in Periodic Functions: Interval helps in identifying specific cycles of periodic functions ensuring accurate descriptions and predictions of the behavior of functions such as sine.

In the given problem, understanding how the interval \([0, \frac{7\pi}{3}]\) interacts with the periodic nature of the sine function is crucial in identifying the unique sine value within that range.