Problem 32 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. If \(\tan \theta+\tan \left(\theta+\frac{\pi}{3}\right)+\tan \left(\theta+\frac{2 \pi}{3}\right)=3\), then \(\theta\) is (a) \((2 n+1) \frac{\pi}{12}\) (b) \(\left(n \pi \pm \frac{\pi}{3}\right)\) (c) \((4 n+1) \frac{\pi}{12}\) (d) None

Step-by-Step Solution

Answer

(d) None of them is the correct answer.

Step 1: Simplify the Given Equation

Start by applying the formula for \(tan(A + B)\) in the equation: \(\tan(\theta) + \tan(\theta + \frac{\pi}{3}) + \tan(\theta + \frac{2\pi}{3}) = 3\). The expression \(tan(\theta + \frac{\pi}{3})\) can be written as \frac{\tan(\theta) + \tan(\frac{\pi}{3})}{1 - \tan(\theta)\tan(\frac{\pi}{3})}\, and \(\tan(\theta + \frac{2\pi}{3})\) can be written as \(-\tan(\frac{\pi}{3} - \theta)\) by using the formula of the tangent of the sum of angles and tangent of the difference between the angles.

Step 2: Calculate the Tangent Values

Now, calculate the values of \(\tan(\frac{\pi}{3})\) and \(\tan(\frac{2\pi}{3})\), which are \(\sqrt{3}\) and -\(\sqrt{3}\) respectively. Substitute these values in the equation, and solve for \(\tan(\theta)\) to get \(\tan(\theta)=1\). The general solution of \(\tan(\theta)=1\), gives \(\theta=\(4n + 1) \frac{\pi}{4}\) where n ∈Z (set of integers).

Step 3: Compare Result with Options

Finally, compare the solution with the given options. As the solution \(\(4n + 1) \frac{\pi}{4}\) does not match any of the options (a), (b), or (c), the correct option should be (d): None.

Key Concepts

Tangent Sum and Difference Identities

Understanding the tangent sum and difference identities is crucial for solving a wide range of trigonometric equations. These identities state how the tangent of the sum or difference of two angles can be expressed in terms of the tangents of the individual angles.

Let's look at the identities:

• The tangent of the sum of two angles, A and B, is given by \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\).
• The tangent of the difference between two angles is similarly \(\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).

Applying these identities helps simplify complex trigonometric expressions and is a stepping stone toward finding solutions to equations like those presented in the exercise.

For example, given \(\tan(\theta + \frac{\pi}{3})\), we can substitute A with \(\theta\) and B with \(\frac{\pi}{3}\) in the sum identity to rewrite the expression appropriately. After finding the values of tangent for pi over three and substituting them back into the equation, we can move towards solving for \(\theta\).

General Solution of Trigonometric Equations

The general solution of trigonometric equations is an essential concept that helps students find all possible solutions to a trigonometric equation, rather than just the principal solution.

For instance, when faced with an equation like \(\tan(\theta) = 1\), we know that one solution is \(\theta = \frac{\pi}{4}\). However, since the tangent function has a period of \(\pi\), the general solution must include all angles that differ from the principal solution by integer multiples of \(\pi\). Specifically, we express the general solution for the tangent as \(\theta = n\pi + \alpha\), where \(\alpha\) is the principal solution, and \(n\) is an integer.

In the given exercise, by finding that \(\tan(\theta) = 1\) and knowing that the principal solution is \(\theta = \frac{\pi}{4}\), the general solution is written as \(\theta = (4n+1)\frac{\pi}{4}\), accounting for all periodic solutions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. These identities are the backbone of solving trigonometric equations and manipulating expressions.

Some fundamental identities include:

• The Pythagorean identities: \(\sin^2 A + \cos^2 A = 1\), \(1 + \tan^2 A = \sec^2 A\), and \(1 + \cot^2 A = \csc^2 A\).
• The reciprocal identities: \(\sin A = \frac{1}{\csc A}\), \(\cos A = \frac{1}{\sec A}\), and \(\tan A = \frac{1}{\cot A}\).
• The angle sum and difference identities as previously discussed for tangent, and similar identities for sine and cosine.

These identities allow us to express trigonometric functions in different forms and facilitate the process of solving equations. By knowing these identities, one can approach and solve complex problems systematically, as shown in the step-by-step solution for the exercise.