Problem 5 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(3 \tan \left(\theta-15^{\circ}\right)=\tan \left(\theta+15^{\circ}\right)\)

Step-by-Step Solution

Answer

The solution is \(\theta = \frac{1}{2}\tan^{-1}(4\tan(15))\)

Step 1 - Express the equation using Tangent Addition Formula

Using tangent addition formulas, the given equation is rewritten as, \[3\frac{\tan(\theta) - \tan(15)}{1 + \tan(\theta)\tan(15)} = \frac{\tan(\theta) + \tan(15)}{1 - \tan(\theta)\tan(15)}\]

Step 2 - Simplify the Equation

The equation can be simplified further to obtain, \[3(\tan(\theta) - \tan(15)) (1 - \tan(\theta)\tan(15)) = (\tan(\theta) + \tan(15))(1 + \tan(\theta)\tan(15))\]

Step 3 - Cancel the common Factors

The equation can be simplified by cancelling common factors on both sides and rearranging to obtain, \[2\tan(\theta) = 4\tan(15)\]

Step 4 - Find the Value of Theta

Finally, the value of \(\theta\) can be obtained by isolating \(\theta\) from the equation. Thus, \[\theta = \frac{1}{2}\tan^{-1}(4\tan(15))\]

Key Concepts

Tangent Addition Formula

The tangent addition formula is pivotal in solving our equation, and it provides a way to combine tangents of two angles. The formula is given by \[\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}\]and similarly for subtraction, \[\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}\].
It's essentially a tool from trigonometry that simplifies the process of adding or subtracting two angles before finding their tangent. This formula is derived from the sine and cosine addition formulas, using the identity \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\].
The application of this formula is what enabled the step-by-step solution to transition from the initial problem into a more tractable equation. Without utilizing this identity, the problem involving \(\tan(\theta - 15^\circ)\) and \(\tan(\theta + 15^\circ)\) would remain complex and hard to solve.

Solving Trigonometric Equations

Trigonometric equations can often look intimidating, but they are simply equations involving trigonometric functions like sine, cosine, and tangent that need to be solved for angles, typically denoted as \(\theta\).
To solve them, one often goes through a process of manipulating the equation using trigonometric identities to simplify it to a form where the angle can be isolated and calculated. The steps involve:

• Applying appropriate trigonometric identities or formulas.
• Simplifying the equation to its basic form.
• Isolating the variable (the angle) on one side of the equation.
• Using inverse trigonometric functions to find the angle.

In our exercise, these steps are followed to transform the original complex equation into a simpler one, eventually isolating \(\theta\) and finding its value using the inverse tangent function, \(\tan^{-1}\). It's crucial to recognize which identities apply and how to manipulate the equation without making algebraic errors. This is an essential part of a student's mathematical toolkit in solving problems involving periodic phenomena or wave patterns.

Trigonometric Identities

Trigonometric identities are equations that are true for all possible values of the variable involved. They are the backbone of simplifying and solving trigonometric equations. Three major types of identities are:

• Pythagorean identities, such as \(\sin^2(\theta) + \cos^2(\theta) = 1\).
• Reciprocal identities, like \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
• Angle sum and difference identities, which include the tangent addition formula used in our problem.

These identities are not just mathematical curiosities; they are profoundly useful in various branches of science and engineering. In the context of our exercise, these identities were instrumental in manipulating the equation to simplify it to a point where \(\theta\) can be isolated. Understanding these identities allows us to effectively navigate through complicated trigonometric expressions by revealing underlying simplicities.