Problem 13 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The number of points of intersection of \(2 y=1\) and \(y=\cos x\) in \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) is (a) 1 (b) 2 (c) 3 (d) 4

Step-by-Step Solution

Answer

(b) 2

Step 1: Simplify Equation

The first step is simply to simplify the equation \(2 y=1\) to \(y=0.5\).

Step 2: Identify Range of Cosine Function

The cosine function oscillates between -1 and 1, inclusive. Therefore, \(y=0.5\) is a valid point within the range of the cosine function.

Step 3: Find Intersection Points

To find the points of intersection, the value of x for which \(\cos x = 0.5\) need to be determined within the interval \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\). Looking at the unit circle or a cosine function graph, it is clear that there are two such points: \(x=\frac{\pi}{3}\) and \(x = -\frac{\pi}{3}\) in the provided interval.

Key Concepts

Solving Trigonometric Equations

The process of finding the values of variables that satisfy a trigonometric equation is known as solving trigonometric equations. These equations involve trigonometric functions such as sine, cosine, and tangent. The solving process often includes a few key steps, such as simplifying the equation, identifying the function's period, and determining the specific range or interval where the solution is valid.

For instance, if the given equation is \(2y = 1\), which simplifies to \(y = 0.5\), and we are told to find the values of \(x\) for which \(y = \text{cos}(x)\), we look for all \(x\) within the specified interval where the cosine of \(x\) equals 0.5. Since trigonometric functions are periodic, there may be multiple solutions, often repeating every \(2\text{π}\) for sine and cosine functions. Knowing the principal domain for these functions, the solutions can be pinpointed accurately by using trigonometric identities, inverse functions, and reference angles.

Range of Cosine Function

The cosine function, one of the basic trigonometric functions, oscillates between -1 and 1. This is known as the range of cosine function, and it means that for any angle \(x\), the value of \(\text{cos}(x)\) will be within that interval. No matter the size of \(x\), the resulting cosine value is never less than -1 or greater than 1.

Understanding the range is crucial when you're solving equations where the cosine function is set equal to a certain value, as in our exercise \(y = \text{cos}(x) = 0.5\). Here, since 0.5 is within the range of the cosine function, this tells us that there are angles for which the cosine value is indeed 0.5. We can be certain of finding solutions within the specified domain because the specified value of 0.5 does not exceed the range limits of the cosine function.

Unit Circle

The unit circle is a fundamental tool in trigonometry, consisting of a circle with a radius of 1, centered at the origin of a coordinate system. This circle helps to define the sine and cosine functions for all angles. On the unit circle, any point along the circumference can be described by the coordinates \((\text{cos}(x), \text{sin}(x))\), with \(x\) being the angle formed by the line connecting the origin to that point and the positive x-axis.

When you're seeking to find the points of intersection between \(y=0.5\) and \(y=\text{cos}(x)\), as in the given exercise, the unit circle shows that there are indeed two angles where the cosine value is 0.5 within the specified interval of \(-\frac{\text{π}}{2} \text{leq} x \text{leq} \frac{\text{π}}{2}\). These are \(x=\frac{\text{π}}{3}\) and \(x=-\frac{\text{π}}{3}\), as seen where the horizontal line \(y=0.5\) intersects the circle. Understanding the unit circle allows students to visualize these solutions and grasp why there are two points of intersection rather than one, three, or four, thereby answering the exercise with the correct choice (b) 2.