Problem 34 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation \(3^{\sin 2 x+2 \cos ^{2} x}+3^{1-\sin 2 x+2 \sin ^{2} x}=28\) is satis- fied for the values of \(x\) given by (A) \(\cos x=0\) (B) \(\tan x=-1\) (C) \(\tan x=1\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (C) \(\tan x = 1\).

Step 1: Understand the Equation

The equation given is a sum of two exponential terms: \(3^{\sin 2x + 2 \cos^2 x} + 3^{1 - \sin 2x + 2 \sin^2 x} = 28\). Identify that the solution involves trigonometric identities and exponential equations. The aim is to find values of \(x\) for which the equation holds true.

Step 2: Simplify Using Trigonometric Identities

Use trigonometric identities to simplify the equation. Recognize that \(\sin 2x = 2 \sin x \cos x\). This might help in expressing \(\cos^2 x\) and \(\sin^2 x\) in terms of \(\tan x\). Use \(\sin^2 x = \frac{1}{1+\tan^2 x}\) and \(\cos^2 x = \frac{\tan^2 x}{1+ \tan^2 x}\) to rewrite the equation.

Step 3: Test Specific Values for \(x\)

Given the options, substitute specific values \(\tan x = -1\), \(\tan x = 1\), and \(\cos x = 0\). Start with \(\tan x = 1\), which implies \(\sin x = \cos x\) when \(x = \frac{\pi}{4} + n\pi\). Substitute this into the equation and check if it satisfies the equation.

Step 4: Verification for \(\tan x = 1\)

For \(x = \frac{\pi}{4}\), \(\sin x = \cos x = \frac{\sqrt{2}}{2}\). Calculate \(\sin 2x = 1\). Plug in these values into the original equation and ensure it simplifies down to 28 to verify correctness.

Step 5: Confirm by Substituting \(\tan x = -1\)

Repeat the same process for \(\tan x = -1\). Use the identity \(\tan x = \frac{\sin x}{\cos x} = -1\) which implies \(\sin x = -\cos x\). Use these in the equation to check if it balances out.

Step 6: Conclusion

Compare the results of substitution from Step 4 and Step 5. If Step 4 balances correctly while Step 5 doesn't, it confirms that \(\tan x = 1\) satisfies the equation. Hence, Option C is correct.

Key Concepts

Exponential Equations

Exponential equations are equations where the variables appear as exponents. In our context, the equation is given with base 3, involving terms like \(3^{\sin 2x + 2 \cos^2 x}\). When dealing with exponential equations, it's important to understand the properties of exponents, as they can greatly affect the techniques you use to solve them.
For instance, knowing that \(a^{m+n} = a^m \times a^n\) allows you to handle the expressions methodically.

• Identify the base and try to simplify the exponent.
• Look for methods to equalize or substitute to simplify the equation.

Being able to express terms using trigonometric identities, as seen in this exercise, helps break down complex exponents.

Trigonometric Identities

Trigonometric identities are powerful tools for solving equations involving trigonometric functions. In this problem, we used the identity \(\sin 2x = 2 \sin x \cos x\). These identities relate the functions in a way that allows us to simplify expressions.Another key identity is the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Using these, we can express \(\sin^2 x\) and \(\cos^2 x\) in terms of \(\tan x\), which can be very helpful:

• \(\sin^2 x = \frac{1}{1+\tan^2 x}\)
• \(\cos^2 x = \frac{\tan^2 x}{1+ \tan^2 x}\)

These simplifications are crucial in breaking down the equation into a more manageable form that can then be solved by testing specific values for \(x\).

Tan Function

The tan function, often denoted as \(\tan x\), is the ratio of \(\sin x\) to \(\cos x\). This means \(\tan x = \frac{\sin x}{\cos x}\).
In trigonometric equations, especially those involving expressions like \(\tan x = 1\) or \(\tan x = -1\), you can infer a lot about the angle \(x\).

• For \(\tan x = 1\), \(x\) takes values like \(\frac{\pi}{4} + n\pi\).
• For \(\tan x = -1\), \(x\) is like \(\frac{3\pi}{4} + n\pi\).

Understanding these standard angles allows us to test possibilities in equations efficiently. Knowing the behavior of \(\tan x\) helps expand solutions using familiar reference angles.

Sin Function

The sin function, denoted as \(\sin x\), is one of the primary trigonometric functions and is used to describe the y-coordinate of a point on the unit circle. It is cyclic with a period of \(2\pi\). This characteristic makes it very useful in repeating pattern problems, like those involving exponential equations.

• \(\sin 2x\) doubles the angle, causing the function to complete two cycles over \(0\) to \(2\pi\).
• \(\sin x\) ranges from -1 to 1.

When \(\sin x = \cos x\), the angle is \(\frac{\pi}{4}\) or similar, allowing solution testing for trigonometric problems. Knowing how \(\sin\) interacts with other functions makes solving equations straightforward when expressions like \(\sin 2x\) are involved.