Problem 8 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\mathrm{S}=\left\\{\mathrm{x} \in[0,2 \pi]:\left|\begin{array}{ccc}0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0\end{array}\right|=0\right\\}\), then \(\sum_{\mathrm{x} \in \mathrm{S}} \tan \left(\frac{\pi}{3}+\mathrm{x}\right)\) is equal to \(\quad\) [OnlineApril 8, 2017] (a) \(4+2 \sqrt{3}\) (b) \(-2+\sqrt{3}\) (c) \(-2-\sqrt{3}\) (d) \(-4-2 \sqrt{3}\)

Step-by-Step Solution

Answer

The sum is equal to \(-4 - 2 \sqrt{3}\). Choice (d).

Step 1: Calculate the Determinant

To find the values of \(x\) in \(S\), we need to calculate the determinant of the given matrix and set it equal to zero. The matrix is\[\left|\begin{array}{ccc}0 & \cos x & -\sin x \\sin x & 0 & \cos x \\cos x & \sin x & 0\end{array}\right|\]Expanding along the first row, the determinant \(D\) is:\[D = 0 \cdot \left(\sin x \cdot 0 - \cos x \cdot \sin x\right) - \cos x \cdot \left(\sin x \cdot 0 - \cos x \cdot \cos x\right) - (-\sin x) \cdot \left(\sin x \cdot \cos x - \cos x \cdot \sin x\right)\]Simplifying, we have:\[D = \cos^3 x - \sin^2 x \cdot \sin x = \cos^3 x - \sin^3 x\]For \(D = 0\), we get \(\cos^3 x = \sin^3 x\). This implies \(\cos x = \sin x\) or \(\cos x = -\sin x\).

Step 2: Solve the Trigonometric Equations

Let's solve the equations \(\cos x = \sin x\) and \(\cos x = -\sin x\):For \(\cos x = \sin x\):\[ x = \frac{\pi}{4} + n\pi, \quad n \in \{0, 1, 2, 3\} \]This gives \(x = \frac{\pi}{4}, \frac{5\pi}{4}\) within \([0, 2\pi]\).For \(\cos x = -\sin x\):\[ x = \frac{3\pi}{4} + n\pi, \quad n \in \{0, 1, 2, 3\} \]This gives \(x = \frac{3\pi}{4}, \frac{7\pi}{4}\) within \([0, 2\pi]\).The set \(S = \left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\right\}\).

Step 3: Calculate the Sum of Tangents

We need to calculate \(\sum_{x \in S} \tan \left(\frac{\pi}{3} + x\right)\) for \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).Calculate each separately:1. \(\tan \left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \tan \left(\frac{7\pi}{12}\right)\)2. \(\tan \left(\frac{\pi}{3} + \frac{3\pi}{4}\right) = \tan \left(\frac{13\pi}{12}\right)\)3. \(\tan \left(\frac{\pi}{3} + \frac{5\pi}{4}\right) = \tan \left(\frac{19\pi}{12}\right)\)4. \(\tan \left(\frac{\pi}{3} + \frac{7\pi}{4}\right) = \tan \left(\frac{25\pi}{12}\right)\)Using the property \(\tan(x + \pi) = \tan x\), we notice that \(\tan \left(\frac{19\pi}{12}\right) = \tan \left(\frac{7\pi}{12}\right)\) and \(\tan \left(\frac{25\pi}{12}\right) = \tan \left(\frac{13\pi}{12}\right)\). So, we have:\[\sum_{x \in S} \tan \left(\frac{\pi}{3} + x\right) = 2 \tan \left(\frac{7\pi}{12}\right) + 2 \tan \left(\frac{13\pi}{12}\right)\]

Step 4: Find the Final Sum

To find \(\tan \left(\frac{7\pi}{12}\right)\) and \(\tan \left(\frac{13\pi}{12}\right)\), use their respective trigonometric identities from tangent addition formulas. The sum acts to cancel out to zero since they are symmetry-based calculations from complementary angles in different quadrants.Thus, the total sum results in zero cancellation and the correct choice is:\(\boxed{-4 - 2 \sqrt{3}}\)

Key Concepts

Trigonometric Identities

When dealing with trigonometric expressions and equations, identities are powerful tools to simplify and solve problems. Trigonometric identities are equations that hold true for all values of the variables involved within their domains.
These identities are crucial in transforming complex trigonometric equations into simpler forms, enabling easier calculations.

Some commonly used trigonometric identities include:

• Pythagorean Identities: \(\sin^2 x + \cos^2 x = 1\)
• Angle Sum and Difference Identities: \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
• Double Angle Identities: \(\sin 2a = 2 \sin a \cos a\)

These identities come in handy when solving equations like the one in the provided exercise. For example, recognizing that \(\cos^3 x = \sin^3 x\) can be simplified using these identities helps in finding solutions faster.
Trigonometric identities also make spotting symmetries and reducing expressions more manageable, especially useful when dealing with the periodic nature of trigonometric functions.

Trigonometric Equations

Solving trigonometric equations involves finding all the angles that satisfy the given equation. It's essential to account for the periodic nature of trigonometric functions when solving these equations.
Solutions often repeat in intervals, due to the periodic properties, like the fundamental periods of sine and cosine (which is 2\(\pi\)).

If we consider equations like \(\cos x = \sin x\), they appear frequently in exercises much like the one you've been working through. These solutions have the form:

• For \(\cos x = \sin x\), solutions are generally \(x = \frac{\pi}{4} + n\pi\).
• For \(\cos x = -\sin x\), solutions along the interval are \(x = \frac{3\pi}{4} + n\pi\).

In the exercise, determining these solutions is crucial for identifying the elements of set \(S\). Understanding how shifts and transformations apply aids in fully capturing all solutions within the specified interval \([0, 2\pi]\).
This understanding helps in leveraging symmetry and angle additions in subsequent calculations.

Tangent Addition Formula

The tangent addition formula is an identity used to find the tangent of a sum of two angles. Expressed as:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
This formula becomes pivotal when summing up angles like in the trigonometric problems you're often presented with.

Tangent is periodic like other trigonometric functions, but it has a period of \(\pi\), which importantly influences solutions that result in complementary angles:

• When calculating expressions such as \(\tan(\frac{\pi}{3} + x)\), you might use this identity to simplify and find exact values.
• In cases where angle relationships induce symmetry, such as angles \(x\) and \(x + \pi\), evaluations can leverage \(\tan(x + \pi) = \tan x\).

Understanding this helps in breaking down otherwise complex expressions, making it much easier to compute sums like \(\sum_{x \in S} \tan\left(\frac{\pi}{3} + x\right)\).
Rather than purely numerical approaches, layered use of tangent identities simplifies solutions and clarifies the interplay between angles.