Problem 342 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ |\cos x+\sec x| \geq 2 $$

Step-by-Step Solution

Answer

The solution to the inequality \( |\cos x + \sec x| \geq 2 \) is \( x = n\pi \), where \( n \) is an integer.

Step 1: Express Secant In Terms of Cosine

Recognize that secant of an angle equals the reciprocal of the cosine of that angle. Therefore, the inequality will be written as \( |\cos x+1/\cos x| \geq 2 \), where \( \cos x \neq 0 \)

Step 2: Manipulation of Inequality

Knowing that for an inequality \( |a| \geq b \), then \( a \geq b \) or \( a \leq -b \), we set \( a = \cos x + 1/\cos x \) and \( b = 2 \). So, we now write two separate inequalities giving \( \cos x + 1/\cos x \geq 2 \) and \( \cos x + 1/\cos x \leq -2 \)

Step 3: Solve the Two Separate Inequalities

Multiplying both sides of the first inequality by \( \cos x \) (also noting that \( \cos x > 0 \)), yields \( \cos^2 x + 1 \geq 2\cos x \). Rearranging terms, gives \( \cos^2 x - 2 \cos x + 1 \geq 0 \). This simplifies to \( (\cos x -1)^2 \geq 0 \). Since squared quantities are always positive, the solutions are \( x = 2n \pi \) where \( n \) is an integer, because cosine function is always 1 at multiples of \( 2 \pi \). Similarly, solving the second inequality, gives \( x = (2n+1) \pi \) where \( n \) is an integer

Step 4: State Final Solution

From Step 3, you combine the results to give final solution as \( x = n\pi \) for \( n \) an integer

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the core trigonometric functions. It represents the x-coordinate of a point on the unit circle at an angle \( x \) measured from the positive x-axis. For any angle \( x \), \( \cos x \) can take values between -1 and 1.

Some key characteristics of the cosine function include:

• Periodicity: \( \cos x \) has a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) radians.
• Symmetry: The cosine function is even, which implies \( \cos(-x) = \cos x \).
• Key Angles: At 0 and multiples of \( 2\pi \), \( \cos x \) equals 1. At odd multiples of \( \pi \), \( \cos x \) equals -1.

Understanding these properties is crucial when examining trigonometric equations or inequalities, such as in the step-by-step problem solution above. Recognizing where \( \cos x \) reaches specific values helps solve equalities like \( \cos x = 1 \) or \( \cos x = -1 \), both of which are pivotal to finding solutions to trigonometric inequalities.

Secant Function

The secant function, denoted as \( \sec x \), complements the cosine function by serving as its reciprocal: \( \sec x = \frac{1}{\cos x} \). This relationship means that secant is undefined whenever \( \cos x = 0 \), as division by zero is impossible.

Here are some important facts about the secant function:

• Domains and Undefined Points: Secant is undefined at odd multiples of \( \frac{\pi}{2} \) because these are the points where \( \cos x = 0 \).
• Range: Since \( \sec x \) is the reciprocal of \( \cos x \), its range is \(( -\infty, -1 ] \cup [ 1, \infty )\).
• Periodicity: Like cosine, the secant function has a periodicity of \( 2\pi \).

The secant function's relationship with cosine is particularly useful in expressing inequalities, as seen in the provided solution. By re-writing secant in terms of cosine, it becomes easier to handle these functions mathematically, especially when involving inequalities.

Inequality Manipulation

Inequality manipulation is a critical skill in solving mathematical problems, as it allows one to reshape inequalities into solvable forms.

Let's look at some basic principles:

• Property of Absolute Values: For any inequality \( |a| \geq b \), this is equivalent to \( a \geq b \) or \( a \leq -b \). This symmetry allows us to break down complex absolute inequalities into simpler ones.
• Multiplication and Division: When manipulating inequalities, one must be cautious of the sign of the multiplying or dividing term. Multiplying or dividing both sides of an inequality by a negative number reverses the inequality's direction.
• Rewriting Expressions: As demonstrated in the solution, substituting secant with cosine's reciprocal can transform and simplify the inequality.

These techniques are invaluable when working with trigonometric inequalities. In our problem, rewriting and breaking down the initial absolute value into manageable parts was key to finding solutions. The careful treatment of inequalities allows for methodical solution processes in complex trigonometric contexts.