Problem 32 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec} x=7\) and \(\sin 2 x=a-b \sqrt{7}\), then (A) \(a=8\) (B) \(b=22\) (C) \(a=22\) (D) \(b=8\)

Step-by-Step Solution

Answer

None of the options correctly match; closest is (A) \( a=8 \).

Step 1: Understanding the Equation

We have the equation \( \sin x + \cos x + \tan x + \cot x + \sec x + \operatorname{cosec} x = 7 \). Each term is a trigonometric function of \( x \). Our goal is to utilize trigonometric identities to make this equation easier to handle.

Step 2: Exploring Trigonometric Functions

Each trigonometric function involves \( \sin x \) and \( \cos x \). By identities: \( \tan x = \frac{\sin x}{\cos x} \), \( \cot x = \frac{\cos x}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), and \( \operatorname{cosec} x = \frac{1}{\sin x} \).

Step 3: Simplify Using Identities

Using \( \tan x \cdot \cot x = 1 \), \( \sec x \cdot \operatorname{cosec} x = \frac{1}{\sin x \cos x} \) and adding them up: \[ \sin x + \cos x + 1 + 1 + \frac{1}{\cos x} + \frac{1}{\sin x} = 7 \], simplify to \[ \sin x + \cos x + 2 + \frac{\sin x + \cos x}{\sin x \cos x} = 7 \].

Step 4: Setting Simplified Equation

Assume \( a + b = \sin x + \cos x \) then \( a + b + \frac{a + b}{ab} = 5 \). Let \( ab = \frac{1}{2} \). Replacing and solving, \( a + b = 2 \), compute for \( a = b \). We can let \( a = b = 1 \).

Step 5: Find \( \sin 2x \)

Since \( \sin x = \cos x = \frac{1}{\sqrt{2}} \), use the double angle formula \( \sin 2x = 2 \sin x \cos x \). Substitute \( \sin x = \cos x = \frac{1}{\sqrt{2}} \) into the formula to get \( \sin 2x = 1 \).

Step 6: Compare \( \sin 2x \) with the Given Expression

Given \( \sin 2x = a - b\sqrt{7} = 1 \), equate to simplify and solve for \( a \) and \( b\). Ensure \( b \sqrt{7} = 0\) thus \( b = 0\), solve \( a = 1 \).

Step 7: Recall Options and Conclusion

Given options (A) \( a = 8 \), (B) \( b = 22 \), (C) \( a = 22 \), (D) \( b = 8 \), none match with \( a = 1 \) or \( b = 0 \). Option (A) with\( a = 8 \) is closest and possibly considering errors.

Key Concepts

Trigonometric functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in geometry and trigonometry. These functions are particularly useful in calculating unknown angles and sides of right triangles. The most commonly used trigonometric functions include:

• Sine \( (\sin x) \): This function gives the ratio of the length of the side opposite to the angle (opposite side) to the hypotenuse.
• Cosine \( (\cos x) \): This is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent \( (\tan x) \): This ratio is obtained by dividing the sine of the angle by the cosine of the angle. It can also be expressed as the ratio of the opposite side to the adjacent side.
• Cotangent \( (\cot x) \): This function is the reciprocal of the tangent, or the ratio of the adjacent side to the opposite side.
• Secant \( (\sec x) \): This is the reciprocal of the cosine function, equal to the ratio of the hypotenuse to the adjacent side.
• Cosecant \( (\operatorname{cosec} x) \): The reciprocal of the sine function, equal to the ratio of the hypotenuse to the opposite side.

Trigonometric functions are widely used in various fields such as physics, engineering, and even in signal processing.

Double angle formula

The double angle formula is a trigonometric identity used to find the values of trigonometric functions of double angles. It is particularly useful in simplifying expressions involving angles that are multiples of other angles. For the sine, cosine, and tangent functions, the double angle formulas are as follows:

• Sine: \[ \sin 2x = 2 \sin x \cos x \]
• Cosine: \[ \cos 2x = \cos^2 x - \sin^2 x \]
• Tangent: \[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]

In the given exercise, we specifically focus on the sine double angle formula. By substituting known values of sine and cosine into this formula, we can solve for expressions involving double angles.
It is crucial to understand these formulas and how they can be used in different contexts, such as solving equations like the one in the exercise. Mastery of these identities allows for efficient problem-solving in various mathematical scenarios.

Simplifying equations

Simplifying equations is a fundamental skill in mathematics that helps make complex problems more manageable. It often involves reducing expressions to their simplest form by using mathematical identities or combining like terms. In the context of trigonometry, simplification might involve:

• Using trigonometric identities to substitute equivalent expressions (e.g., using \( \tan x = \frac{\sin x}{\cos x} \)).
• Adding and factoring to group terms effectively, helping to reveal simplified relations.
• Applying recognizable patterns or techniques such as the distributive property or combining similar terms.

In the exercise provided, simplifying the equation was key in finding the solution. By recognizing and utilizing various trigonometric identities, such as those for \( \tan x \), \( \cot x \), \( \sec x \), and \( \operatorname{cosec} x \), the equation could be reduced to its simplest terms. This process highlighted the potential values for sine and cosine needed to satisfy the overall equation.
By simplifying, not only is the problem made more accessible, but it also prepares the mathematical groundwork for more advanced topics, ensuring a deeper understanding of the core concepts involved.