Problem 65 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The general solution of the equation \(2^{\cos ^{2} x}+1=3.2^{-\sin ^{2} x}\) is (a) \(n \pi \cdot n \in z\) (b) \((n+1) \pi, n \in z\) (c) \((n-1) \pi, n \in z\) (d) None

Step-by-Step Solution

Answer

The solution to the given equation is (a) \(n \pi\), where \(n\) is any integer.

Step 1: Substitute the Trigonometric Identity

Replace \(\cos ^{2} x\) with \(1-\sin^2 x\) and \(\sin ^{2} x\) with \(1-\cos^2 x\). Then the equation becomes \(2^{1-\sin^2 x}+1=3.2^{1-\cos^2 x}\).

Step 2: Simplify the Equation

We simplify the equation by subtracting 1 from both sides, which yields \(2^{1-\sin^2 x}=3.2^{1-\cos^2 x}-1\). But as we know, \(2^1 = 2\) and \(2^{-\sin^2 x}=2^{-1}\) if \(\sin^2 x = 1\). And the same applies to the right side of the equation. So, if \(\sin^2 x = 1\) or \(\cos^2 x = 1\), both sides of the equation become equal. By the definitions of trigonometric functions, \(\sin^2 x = 1\) when \(x = (2n+1)\pi/2\) and \(\cos^2 x = 1\) when \(x = n\pi\).

Step 3: Find the Solution

So the solutions are those values of x for which either \(\sin^2 x = 1\) or \(\cos^2 x = 1\). So \(x = (2n+1)\pi/2\) or \(x = n\pi\). Putting these two conditions together, we get all integer multiples of \(\pi\) as solutions. Hence \(n\pi\), where \(n\) is an integer, is the general solution.

Key Concepts

General Solution

In trigonometry, finding a general solution means identifying all possible solutions to a trigonometric equation. These solutions are expressed in terms of a constant like \( n \) which represents an integer. When solving trigonometric equations, the periodic nature of these functions allows them to have an infinite number of solutions, distributed over regular intervals. For instance, in the given equation, the solutions are given as \( n\pi \) where \( n \) is any integer. This implies that no matter how far along the number line you go, you can keep finding solutions at regular intervals defined by \( \pi \). Being able to express these solutions generally rather than individually is a powerful tool in dealing with trigonometric equations.

Trigonometric Identities

Tucked effortlessly within many trigonometric equations are identities, which are equalities involving trigonometric functions that are true for any value of the involved variables. One remarkable trigonometric identity is \( \sin^2 x + \cos^2 x = 1 \). In solving the provided problem, identifying and employing such an identity helps in simplifying the equation. By substituting \( \cos^2 x = 1 - \sin^2 x \) and \( \sin^2 x = 1 - \cos^2 x \), the equation becomes more manageable. These substitutions and transformations enable us to simplify complex trigonometric expressions and make it possible to confidently work towards a general solution.

Cosine Function

The cosine function, \( \cos x \), is a fundamental trigonometric function that is associated with the adjacent side and hypotenuse in a right-angled triangle. The function has a periodicity of \(2\pi\), meaning it repeats every \(2\pi\) radians. In the context of the exercise, \( \cos^2 x = 1 \) at points where \( x = n\pi \), illustrating the cosine function's periodicity and symmetry about the x-axis. This quality makes it instrumental in providing solutions to equations such as the one given, where these repeated cycles play a crucial role in forming a complete set of general solutions.

Sine Function

Similar to the cosine function, the sine function \( \sin x \) is of great importance in trigonometry. The sine function gives the ratio of the opposite side to the hypotenuse in a right-angle triangle and has a periodicity of \(2\pi\). When we talk about \( \sin^2 x = 1 \), this occurs at points like \( x = (2n+1)\pi/2 \), echoing the sine function's behavior at these critical points. The fact that these points appear at specific intervals reinforces the capability of sine to describe periodic cycles. This property is utilized in solving equations by anticipating and identifying these periodic solutions throughout the domain.