If \(\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=[(2 \pi) / 3]\) then \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\) (a) \((\pi / 3)\) (b) \((5 \pi / 6)\) (c) \((\pi / 2)\) (d) \((3 \pi / 2)\)

Understanding the properties of inverse trigonometric functions is crucial when tackling problems in JEE Maths.

One fundamental property is the relationship between the sine and cosine functions: for any angle \(a\), the sum of its sine inverse and cosine inverse is always \(\frac{\pi}{2}\), expressed as \(\sin^{-1}a + \cos^{-1}a = \frac{\pi}{2}\).

Another essential aspect is the domain and range of these functions. For instance, the inverse sine function, \(\sin^{-1}\), is only defined for values between -1 and 1, and its range is \(\Big[ -\frac{\pi}{2}, \frac{\pi}{2} \Big]\).

Let's not forget about the principal values. The concept of principal values is integral to understanding inverse trigonometry. These values restrict the range of the functions to prevent ambiguity, as the trigonometric functions are not one-to-one.

To ensure effectiveness in JEE preparation, let's list some of the properties using bullet points:

• Sum of \(\sin^{-1}\) and \(\cos^{-1}\) for the same angle \(a\) is \(\frac{\pi}{2}\).
• Domains of \(\sin^{-1}\) and \(\cos^{-1}\) are between -1 and 1.
• The ranges for \(\sin^{-1}\) and \(\cos^{-1}\) are \(\Big[-\frac{\pi}{2}, \frac{\pi}{2}\Big]\) and \(\Big[0, \pi\Big]\), respectively.
• They have principal values to maintain function uniqueness.
• Adding or subtracting \(\pi\) to angles in inverse trigonometric functions can yield results in different quadrants, another important consideration for solutions.

Familiarity with these properties can greatly simplify the process of solving problems involving inverse trigonometric functions.

Trigonometric equations can often seem daunting, but with a systematic approach and understanding of inverse functions, they can be approached more confidently.

When solving trigonometric equations, it's important to start by isolating the trigonometric function. Sometimes this involves using algebraic manipulations or trigonometric identities to simplify the equation. Once the function is isolated, it can be converted into its inverse to find the specific angle or solution. However, always remember to consider the domain and range of the inverse functions, as mentioned earlier.

In case of multiple angles or a range of solutions, it's essential to determine all possible angles that satisfy the equation within the given interval, which could be crucial for JEE problems that look for a set of solutions.

To guide students through these steps, we can list a few key tips:

• Isolate the trigonometric function from the equation before attempting to solve.
• Use known identities and properties to simplify equations whenever possible.
• Apply inverse trigonometric functions carefully, remembering their limited domains and ranges.
• Consider all possible angles that might be solutions, especially when solving systemic problems like those found in the JEE.

Following these pointers can help students make significant progress in their understanding and ability to solve trigonometric equations.

The Joint Entrance Examination (JEE) is known for its challenging maths problems, which often require a deep conceptual understanding and problem-solving skills.

JEE maths problems, such as those involving inverse trigonometric functions, demand precision and attention to detail. These problems are designed to test a student's ability to apply concepts in various situations, often combining multiple mathematical principles.

In the case of our example problem involving the sum of inverse trigonometric functions, a thorough step-by-step approach was necessary. However, it's also important for students to cross-verify their answers with the given options. If there is a discrepancy, students should reassess their steps and ensure that all properties and identities are applied correctly.

For successful resolution of JEE math problems, consider these strategies:

• Read the problem statement carefully and note any given constraints.
• Diagram the problem if possible, visual aids can provide additional clarity.
• Be methodical in your solution approach, documenting each step clearly.
• Double-check your work against the given options or use an alternative method to verify the answer.

Preparing for JEE maths problems is about consistent practice and reviewing errors to prevent them in the future. By incorporating these strategies into their study habits, students can increase their ability to solve complex problems and perform better in competitive exams.