Problem 23 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\sin \theta+\cos \theta=\frac{\sqrt{7}}{2}\) and \(0<\theta<\pi / 6\), then \(\tan \left(\frac{\theta}{2}\right)\) equals (A) \(\sqrt{7}-2\) (B) \(\frac{1}{3}(\sqrt{7}-2)\) (C) \(2-\sqrt{7}\) (D) \(\frac{1}{3}(2-\sqrt{7})\)

Step-by-Step Solution

Answer

\(\tan \left(\frac{\theta}{2}\right) = \frac{1}{3}(\sqrt{7}-2)\).

Step 1: Using the Identity

Given \( \sin \theta + \cos \theta = \frac{\sqrt{7}}{2} \), we can use the identity: \( \sin \theta + \cos \theta = \sqrt{2} \sin \left(\theta + \frac{\pi}{4}\right) \) to solve for \( \theta \). Note that \( \sqrt{2} \sin \left(\theta + \frac{\pi}{4}\right) = \frac{\sqrt{7}}{2} \).

Step 2: Solve for \(\sin \left(\theta + \frac{\pi}{4}\right)\)

Divide both sides by \(\sqrt{2}\) to isolate the sine function: \( \sin \left(\theta + \frac{\pi}{4}\right) = \frac{\sqrt{7}/2}{\sqrt{2}} = \frac{\sqrt{14}}{4} \).

Step 3: Express \(\tan \frac{\theta}{2}\) in terms of \(\sin \theta\) and \(\cos \theta\)

Using the identity \( \tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta} \), express \(1 - \cos \theta\) and calculate \(\tan \frac{\theta}{2}\) based on given conditions.

Step 4: Use other identities as needed

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \) and \( \cos \theta \) using known trigonometric transformations if necessary.

Step 5: Calculate \(\tan \frac{\theta}{2}\) explicitly

Compute values and simplify to reach one of the given options.

Key Concepts

Trigonometric Identities

In trigonometry, identities are equations that hold true for all values of the variables within their domain. Two key identities often used in problems like this include the Pythagorean identity and the angle sum identities.
The Pythagorean identity states that \[ \sin^2 \theta + \cos^2 \theta = 1 \] This relationship helps us verify or find unknown values of sine or cosine when one is known. In our problem, knowing \( \sin \theta + \cos \theta = \frac{\sqrt{7}}{2} \) aids in setting equations to find other trigonometric functions. Keeping this in mind, another useful segment is the identity related to sum of angles:\[ \sin \theta + \cos \theta = \sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) \] This turns a cumbersome sum into a simpler single sine function, making calculations straightforward with a reduced complexity. Utilize these identities strategically for solving different equations and conversions.

Half-Angle Formulas

The half-angle formulas are helpful trigonometric tools, especially when you need to find the tangent of half an angle. These formulas are derived from the angle sum and double angle identities. For tangent, the half-angle formula is:\[ \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} \] This formula assists in finding the tangent of half the angle directly using sine and cosine values of the full angle \( \theta \).
We use it in problems like the one given by substituting known values, thus simplifying the problem-solving process. Recall that both \( \sin \theta \) and \( \cos \theta \) can be calculated or derived using their respective identities, allowing us to compute \( \tan \left( \frac{\theta}{2} \right) \) accurately. As half-angle identities rely on the relationships between functions, understanding the derivation and application of these formulas is crucial.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate specific terms or make an equation easier to solve. In trigonometry, it is essential for transforming trigonometric expressions and solving complex problems.
Consider the problem of finding \( \tan \left( \frac{\theta}{2} \right) \). Algebraic manipulation allows us to:

• Simplify the equation \( \frac{1 - \cos \theta}{\sin \theta} \)
• Use known identities to substitute and simplify terms
• Reorganize equations to express one variable in terms of others

Proper manipulation requires careful handling of fractions and radicals, as seen when dividing by \( \sqrt{2} \) or finding a common denominator to simplify fractions further. Applying these manipulations turns a complex expression into a solvable form, making them integral to effectively handling trigonometric problems.