Problem 75 - A Complete Resource Book in Mathematics for JEE Main

Question

I. The value of \(\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)\) at \(x=\frac{1}{5}\) is (A) \(\frac{2}{3 \sqrt{5}}\) II. If \(\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1\), then \(x=\) (B) \(-\frac{2 \sqrt{6}}{5}\) III. The value of \(\tan \left\\{\cos ^{-1}\left(-\frac{2}{7}\right)-\frac{\pi}{2}\right\\}\) is (C) \(\frac{3 \pi}{4}\) IV. If \(\sqrt{p}+\cos ^{-1} \sqrt{1-p}+\cos ^{-1} \sqrt{1-q}=\frac{3 \pi}{4}\), then \(q=\) (D) \(\frac{1}{5}\)

Step-by-Step Solution

Answer

The answers are: (A) \(\frac{2}{3 \sqrt{5}}\), (B) \(-\frac{2 \sqrt{6}}{5}\), (C) classic unknown formula translation, (D) \(\frac{1}{5}\).

Step 1: Understanding the Expression for Part I

We need to evaluate \(\cos \left(2 \cos^{-1} x + \sin^{-1} x\right)\) where \(x = \frac{1}{5}\). Since \(\cos^{-1} x\) and \(\sin^{-1} x\) relate to angles in a right triangle where \(x\) is the adjacent side over the hypotenuse for cosine and opposite for sine, we can express these in terms of a common angle and combine.

Step 2: Use the Sine and Cosine Identities

Express \(\cos^{-1} x\) as an angle \(\theta\) such that \(\cos \theta = x\). Then, \(\sin^{-1} x\) can be expressed as \(\phi\) such that \(\sin \phi = x\). Using trigonometric identities, transform and add these angles to evaluate \(\cos(2\theta + \phi)\).

Step 3: Calculate the Angle Expression

Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\) and substituting arrays for \(x = \frac{1}{5}\), we calculate the expression. Use trigonometric angle addition formulas to solve and simplify to find the value of the entire expression.

Step 4: Evaluate Expression for Part II

Given \(\sin(\sin^{-1}(\frac{1}{5}) + \cos^{-1} x) = 1\), explore the possible values of angles since sine reaches 1 at \(\frac{\pi}{2}\). Work out the transformation to determine \(x\) and verify using inverse trigonometric properties.

Step 5: Simplify Expression for Part III

Re-format \(\tan\left\{\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right\}\). The angle transformation using tangent subtraction leads to a clean calculation after reformulation of reference angles.

Step 6: Solve and Simplify for Part IV

The equation \(\sqrt{p} + \cos^{-1} \sqrt{1-p} + \cos^{-1} \sqrt{1-q} = \frac{3\pi}{4}\) hints at specific values for \(\cos^{-1}\). Solve for \(q\) using transformation and solving inverse angles.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions to one another. These identities simplify complex trigonometric expressions and help in solving trigonometric equations. They are essential in understanding relationships in triangles, particularly right-angled triangles.

One of the fundamental trigonometric identities is the Pythagorean identity, which states:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

This identity is crucial in transforming expressions, especially when simplifying expressions involving powers of sine and cosine functions. For example, if you know \( \sin \theta \), you can easily find \( \cos \theta \) using this identity, and vice versa.

Another important identity is the double angle identity, which allows us to express trigonometric functions of double angles in terms of single angles:

\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]

By using trigonometric identities, you can effectively solve for unknown variables or angles in equations. They are handy in simplifying expressions involving inverse trigonometric functions, often used in calculus and geometry problems. Remember, the key to mastering trigonometric identities is practice and understanding how these identities interrelate.

Angle Addition Formulas

Angle addition formulas are a set of trigonometric identities that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine, cosine, and tangent of the angles themselves. These help simplify calculations involving multiple angles.

For cosine, the angle addition formula is:

\[ \cos (A + B) = \cos A \cos B - \sin A \sin B \]

This formula allows us to find the cosine of an angle composed of two other angles. Similarly, for sine:

\[ \sin (A + B) = \sin A \cos B + \cos A \sin B \]

And for tangent, the additive formula is:

\[ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

These angle addition formulas are invaluable when dealing with problems that involve calculating the trigonometric functions of angle expressions like \(2\theta + \phi\) or transforming complex expressions to simpler forms. They are particularly useful in physics, engineering, and geometry, where precise angle calculations are necessary.

Inverse Trigonometric Properties

Inverse trigonometric properties involve functions that reverse the process of regular trigonometric functions, essentially finding the angle given the trigonometric ratio. Inverse trigonometric functions are represented as \( \sin^{-1} \), \( \cos^{-1} \), and \( \tan^{-1} \).

These functions are crucial for solving equations where you need to find angles, particularly in scenarios involving right triangles or in calculus, where it's essential to invert a trigonometric relationship.

The range of inverse functions provides specific angle ranges, ensuring the functions are well-defined. For instance:

• \( \sin^{-1} x \) yields an angle in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• \( \cos^{-1} x \) gives an angle in the range \([0, \pi]\).
• \( \tan^{-1} x \) results in an angle within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

These ranges are essential to maintain the functions' one-to-one nature.

Another helpful feature of inverse trigonometric properties is their use in resolving ambiguous cases in trigonometry problems, such as in determining angle magnitudes or when specifically finding expressions like \( \tan^{-1} \) of combined angles. Practicing with these properties will make solving trigonometric equations more intuitive and faster.