Problem 57 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The most general values of \(x\) for which \(\sin x+\cos x=\min _{a \in R}\left\\{1, a^{2}-4 a+6\right\\}\) are given by (a) \(2 n \pi, n \in N\) (b) \(2 n \pi+\frac{\pi}{2}, n \in N\) (c) \(n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{4}, n \in N\) (d) None

Step-by-Step Solution

Answer

The correct answer is (c).

Step 1: Analyze the Range Of \(sin x + cos x\)

The maximum value of \(sin x\) or \(cos x\) is 1 and the minimum value of \(sin x\) or \(cos x\) is -1. Therefore, the maximum value of \(sin x + cos x\) comes when we add the maximum values, resulting in 2. Similarly, the minimum value of \(sin x + cos x\) comes when we add the two minimum values, resulting in -2.

Step 2: Find the Minimum Of the Function \(a^{2}-4 a+6\)

This function is a quadratic expression. For such expressions, the minimum or maximum can be calculated at its vertex. The x-coordinate of the vertex of a function given in the standard form \(ax^2 + bx + c\) is given by \(-b/(2a)\), and since our function is \(a^{2}-4 a+6\), the minimum value will occur at \(a = -(-4)/(2*1) = 2\). Substituting \(a=2\) into the equation will give \(2^2 - 4*2 + 6 = 2\).

Step 3: Compare Results

The minimum value of the function \(a^{2}-4 a+6\) is 2, which is within the range of \(sin x + cos x\). This means that there must be some value(s) of \(x\) that can satisfy the original equation \(\sin x+\cos x=a^{2}-4 a+6\). Let's try to find the x that will satisfy the \(sin x + cos x = 2\). When we use the identities \(\sin(\pi/4) = cos(\pi/4) = 1/\sqrt{2}\), we find that \(x = n \pi +(-1)^{n} \frac{\pi}{4} -\frac{\pi}{4}\), where \(n \in N\), fulfills the condition when we add \(\sin x\) and \(\cos x\).

Step 4: Select the Correct Option

The answer should be \(x=n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{4}\), where \(n \in N\). Among the options listed, the correct one is (c).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for solving a variety of equations, especially in the realm of trigonometry. These identities express complex relationships between trigonometric functions and allow us to simplify and solve equations involving trigonometric terms.

For instance, well-known identities such as the Pythagorean identities, \(\sin^2(x) + \cos^2(x) = 1\), help us find the value of one trigonometric function when another is known. In the context of the exercise, we utilized the equality \(\sin(\pi/4) = \cos(\pi/4)\) to solve for the values of \(x\) that satisfy the equation \(\sin x + \cos x = 2\), representing a specific case where both sine and cosine have the same value, \(1/\sqrt{2}\).

These identities are not just mathematical curiosities; they are powerful tools that can simplify complex problems and are used extensively in fields such as physics, engineering, and even in competitive exams where a strong grasp of trigonometry is required.

Quadratic Functions

Quadratic functions have the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola, and it can open either upwards or downwards depending on the sign of \(a\).

To find the vertex of the parabola, and thus determine the minimum or maximum value of the function, we use the formula \(-b/(2a)\). This knowledge is crucial when evaluating the minimum or maximum values as seen in our exercise where we calculated the minimum value of the quadratic expression \(a^2 - 4a + 6\).

Quadratic functions are foundational in algebra and are often encountered in applications involving projectile motion, optimization problems, and in designing structures. In educational settings, especially competitive exams, a thorough understanding of quadratic functions enables students to solve a wide range of problems efficiently.

Trigonometry in Competitive Exams

Trigonometry is a significant topic in competitive exams, be it for college entrance tests, standardized tests like the SAT, or international olympiads. The problems in these exams often test a student's ability to understand and manipulate trigonometric functions and identities.

Competitive exams usually require a deep understanding of concepts such as angle measures, conversion between different units (degrees and radians), solving trigonometric equations, and recognizing the graphs of trigonometric functions. The question we've tackled is a typical example of a problem where solving a trigonometric equation could be the deciding factor for success.

Students must not only memorize identities but also learn how to apply them to various scenarios. Practicing problems similar to our exercise helps build the analytical skills necessary to tackle complex problems and excel in competitive exams where time management and accuracy are key.