Problem 34 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The value of \(\cos \left[\frac{1}{2} \cos \left\\{\cos \left(\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)\right)\right\\}\right]\) is (a) \(\frac{3}{16}\) (b) \(\frac{3}{8}\) (c) \(\frac{3}{4}\) (d) \(\frac{3}{2}\)

Step-by-Step Solution

Answer

\[\frac{\sqrt{3}}{2}\]

Step 1: Calculate the value of inverse sine function

First, determine the cosine of the value inside the inverse sine function. Given \(\sin^{-1}(\sqrt{63}/8)\), use the identity that if \(x\) is acute and \(\sin(x) = a/b\) then \(\cos(x) = \sqrt{b^2 - a^2}/b\) to find \(\cos(\sin^{-1}(\sqrt{63}/8)) = \sqrt{8^2 - (\sqrt{63})^2} / 8 = 1/8\).

Step 2: Substitute the cosine value into the outermost cosine function

Substitute the value of \(\cos(\sin^{-1}(\sqrt{63}/8)) = 1/8\) into the outermost cosine function to get \(\cos\left[\frac{1}{2} \cos \left(\frac{1}{8}\right)\right]\). To simplify this further, realize that the inner cosine is approximately equal to 1 for small values, therefore \(\cos(\frac{1}{8})\) can be approximated as 1.

Step 3: Evaluate the final cosine value

Substitute the value of \(\cos (\frac{1}{8}) = 1\) approximated in step 2 into the whole expression to get \(\cos\left[\frac{1}{2} \cdot 1\right] = \cos\left(\frac{1}{2}\right)\). That equals to \(\sqrt{3}/2\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented by the symbol \( \cos \). It relates the angle of a right triangle to the lengths of the adjacent side and the hypotenuse. Specifically, for an angle \( \theta \), the cosine is the ratio of the adjacent side's length to the hypotenuse. Understanding the cosine function is crucial in trigonometry as it applies in various geometric and mathematical contexts.

• The range of the cosine function is between \(-1\) and \(1\).
• Its period is \(2\pi\), which means the values repeat every \(2\pi\) radians.
• The cosine function is even, meaning \( \cos(-\theta) = \cos(\theta) \).

The cosine function is continuous and smooth, making it useful in continuous mathematical models such as waves and oscillations. In equations, it’s combined with other trigonometric functions to solve problems involving angles and length ratios.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex expressions into more manageable forms. This process often uses identities and algebraic manipulation to reduce a problem to a simpler or more familiar form. For example, in the expression given in our original exercise, we start with nested functions that can initially seem daunting.

• First, use inverse trigonometric identities to recognize values like \( \sin^{-1} \) or \( \cos^{-1} \).
• Apply known trigonometric values for specific angles or use approximations when values are small, simplifying where possible.
• Finally, substitute back into your main expression to find a simplified or approximate solution.

Simplifying these kinds of expressions not only helps in solving equations but also in understanding the relationships and interactions between different trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions, valid for any value of the involved variables where the functions are defined. These identities are essential tools for simplifying expressions and solving trigonometric equations. They help connect different trigonometric functions and provide ways to transform and simplify expressions.

• Reciprocal identities: such as \( \csc \theta = \frac{1}{\sin \theta} \).
• Pythagorean identities: like \( \sin^2\theta + \cos^2\theta = 1 \).
• Angle identities: including double-angle or half-angle identities, are useful for reducing expressions.

In our original example, identities are used to isolate the cosine function from inside a sine function through transformation. Recognizing and applying the correct identity can significantly simplify the process and make it easier to obtain the solution.