Problem 43 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The value of the expression \(\cos ^{2}\left(\frac{\pi}{8}\right)+\cos ^{2}\left(\frac{3 \pi}{8}\right)+\cos ^{2}\left(\frac{5 \pi}{8}\right)+\cos ^{2}\left(\frac{7 \pi}{8}\right)\) is (a) rational (b) integral (c) prime (d) composite

Step-by-Step Solution

Answer

The value of the given set of expressions is an Integral Number.

Step 1: Apply the Identity

We start by applying the identity on each of the expression, giving us: \((1+ \cos (\frac{π}{4}) ) / 2 + (1+ \cos (\frac{3π}{4}))/2 + (1+ \cos (\frac{5π}{4}))/2 + (1+ \cos (\frac{7π}{4}))/2 \)

Step 2: Simplify

By applying values of cos function for the given angles, we get: \((1 + 1/√2)/2 + (1 - 1/√2)/2 + (1 - 1/√2)/2 + (1 + 1/√2)/2 = 1 + 0 + 0 + 1 = 2\)

Step 3: Match with Options

Now, we should match our computed result with the options given. It can be clearly seen that the result 2 is an Integral number (option b). Thus the value of the sum of the given expressions is an integral number.

Key Concepts

Cosine Function Properties

The cosine function is a fundamental part of trigonometry and has properties that are essential for solving various mathematical problems. Here are some crucial properties of the cosine function to consider:

• The cosine function, denoted as \text{\( cos \)}, is an even function, meaning that for every angle \( \theta \), \( \cos(\theta) = \cos(-\theta) \).
• It has a period of \( 2\pi \), so \( \cos(\theta) = \cos(\theta + 2k\pi) \) for any integer \( k \).
• The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse.
• For certain standard angles, the values of the cosine function are well known, such as \( \cos(0) = 1 \), \( \cos(\frac{\pi}{2}) = 0 \), and \( \cos(\pi) = -1 \).
• Using cosine function identities, like the double angle identities and the sum-to-product formulas, we can simplify complex trigonometric expressions.

In the context of the given exercise, understanding these properties allows us to simplify the expression and find that each term can be represented using the half-angle identity, where \( \cos^2(\theta) = \frac{1+\cos(2\theta)}{2} \). This identity is particularly useful for simplifying terms involving squared cosine functions and is a key step in finding the solution to the problem.

Trigonometry Problem Solving

Solving trigonometry problems often requires a systematic approach. Here's a general guide to solving them:

• Identify the trigonometric functions involved and any relevant identities or formulas that can be applied.
• Draw diagrams if the problem involves geometric figures, to better visualize relationships between the angles and sides.
• Substitute known values or angles into the trigonometric functions or identities to simplify the expressions.
• If the equation is complex, break it down into smaller, more manageable parts.
• Algebraic manipulation may also be necessary to isolate terms and solve for unknowns.
• Verify your solution by plugging it back into the original equation or by assessing whether it makes sense within the context of the problem.

In the provided exercise, the systematic application of trigonometric identities and the basic properties of cosine allows us to convert the original expression into a form with known values, enabling us to solve the problem step by step. Through this method, the problem, which initially appears complex, becomes manageable and eventually, yields an integral solution.

Trigonometric Expressions Simplification

Simplifying trigonometric expressions is a crucial skill when working with trigonometry. By applying certain techniques, we can reduce expressions to their most manageable form. Here are a few tips:

• Use fundamental identities like the Pythagorean identities (e.g., \( \sin^2(\theta) + \cos^2(\theta) = 1 \)) to transform and reduce expressions.
• Apply angle identities such as double angle or half angle identities to rewrite trigonometric expressions in terms of other angles.
• Sum-to-product and product-to-sum formulas can turn sums and differences of trigonometric functions into products, which are often simpler to work with.
• Factorization can sometimes be used to combine terms or split complex expressions into simpler parts.
• If possible, cancel out common terms to further reduce the complexity of the expression.

In the step-by-step solution provided, the simplification process involved using the half-angle identity to rewrite squared cosine terms and then replacing the angles with their equivalent standard angle values to reach a simplified numerical result. These simplification techniques are central to solving not just the exercise in question but many other trigonometric problems one might encounter in mathematics.