Problem 7 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the simplest value of \(\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right), \forall x \in\left(\frac{1}{2}, 1\right)\)

Step-by-Step Solution

Answer

The simplest value of the expression \(\cos^{-1} x + \cos^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3x^{2}}}{2}\right)\) for \(x \in \left(\frac{1}{2}, 1\right)\) is \(\cos^{-1}(1)\), which is \(0\).

Step 1: Prior knowledge

Remember, the range of \(\cos^{-1} x\) is \([0 , \pi]\) and the range of the cosine function is \([-1 , 1]\). Also, recall this trigonometric identity: \(A + \cos^{-1} B = \cos^{-1} (A \cos B + \sqrt{1 - B^2} \sin A) \) if \(B \cos A - A \sqrt{1 - B^2} \leq 0 \).

Step 2: Apply the identity

Apply the identity where \(A = x\) and \( B = \frac{x}{2}+\frac{\sqrt{3-3x^{2}}}{2}\) to get \(\cos^{-1} x + \cos^{-1} \left(\frac{x}{2}+\frac{\sqrt{3-3x^{2}}}{2}\right) = \cos^{-1} (x \left(\frac{x}{2}+\frac{\sqrt{3-3x^{2}}}{2}\right) + \sqrt{1 - \left(\frac{x}{2}+\frac{\sqrt{3-3x^{2}}}{2}\right)^2} \sin x) \).

Step 3: Simplify the expression

Simplify the expression inside the cosine inverse function to get \(\cos^{-1}(\frac{3x}{4} + \frac{\sqrt{3-3x^2}}{4} + \sqrt{1 - x^2 - 2x\frac{x}{2} - x\sqrt{3-3x^2} + \frac{3x^2}{4}})\).

Step 4: Final simplification

After simplifying the expression inside the cosine inverse function, the entire expression simplifies to \(\cos^{-1}(1)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in solving equations involving angles and lengths in trigonometry. They provide relationships between the trigonometric functions such as sine, cosine, and tangent. Specifically, these identities allow us to transform complex expressions into simpler forms, making it easier to solve trigonometric equations.

In the given exercise, we utilized a specific identity for the addition of two inverse cosine expressions. This identity allows us to express the sum of two inverse cosine functions as a single inverse cosine function:

• \(A + \cos^{-1} B = \cos^{-1} (A \cos B + \sqrt{1 - B^2} \sin A)\)
• Where \(B \cos A - A \sqrt{1 - B^2} \leq 0\)

This identity simplifies the expression by combining the two inverse cosine terms into a more manageable form.

Understanding these identities can greatly simplify complex trigonometric problems, as they often reveal hidden relationships that aren't immediately obvious. They are essential for solving problems in calculus, physics, and engineering, where trigonometric functions frequently appear.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted as \(\cos\). This function relates an angle of a right triangle to the lengths of its adjacent side and hypotenuse. Mathematically, it is expressed as:\(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)

It can be defined for all real numbers using the unit circle, where \(\theta\) is the angle in radians measured from the positive x-axis. Cosine has a periodicity of \(2\pi\), meaning it repeats its values every \(2\pi\) units.

• Range: \([-1, 1]\)
• Commonly used with angles: 0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\)

The inverse cosine function, \(\cos^{-1}\) or \(\text{arccos}\), is used to determine the angle when the value of cosine is given:

• Domain: \([-1, 1]\)
• Range: \([0, \pi]\)

The cosine function plays a crucial role in the original exercise, as we need to consider both the cosine and inverse cosine functions to simplify the given expression.

Domain and Range

Understanding the domain and range of trigonometric functions, especially inverse trigonometric functions, is key to solving equations. The domain of a function is the complete set of possible input values, while the range is the complete set of output values.

For the inverse cosine function \(\cos^{-1}(x)\), the domain is \([-1, 1]\), as cosine values only lie within this interval. Its range is \([0, \pi]\), indicating the angles for which these values are possible. These constraints ensure that \(\cos^{-1}\) always returns a valid output within the specified interval.
Knowing these intervals helps determine whether expressions and equations have valid solutions across the given bounds.

• Domain of \(\cos^{-1}(x)\): Ensures expressions only include values cosine can achieve
• Range of \(\cos^{-1}(x)\): Ensures solutions are within valid angle intervals

In the exercise provided, the condition \(x \in \left(\frac{1}{2}, 1\right)\) is crucial to ensure that the values being used and derived from the cosine and inverse cosine functions remain valid, ultimately allowing the simplification of the trigonometric expression to the value \(\cos^{-1}(1)\).