Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha=\sin ^{-1} \frac{\sqrt{3}}{2}+\sin ^{-1} \frac{1}{3}\) and \(\beta=\cos ^{-1} \frac{\sqrt{3}}{2}+\cos ^{-1} \frac{1}{3}\), then (A) \(\alpha>\beta\) (B) \(\alpha=\beta\) (C) \(\alpha<\beta\) (D) \(\alpha+\beta=2 \pi\)

Step-by-Step Solution

Answer

(C) \(\alpha < \beta\)

Step 1: Set Initial Values for \\(\alpha\\) and \\(\beta\\)

Given that \(\alpha = \sin^{-1} \frac{\sqrt{3}}{2} + \sin^{-1} \frac{1}{3}\) and \(\beta = \cos^{-1} \frac{\sqrt{3}}{2} + \cos^{-1} \frac{1}{3}\), let's first determine \(\sin^{-1} \frac{\sqrt{3}}{2}\), \(\sin^{-1} \frac{1}{3}\), \(\cos^{-1} \frac{\sqrt{3}}{2}\), and \(\cos^{-1} \frac{1}{3}\).

Step 2: Calculate Inverse Sine Values

\(\sin^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{3}\) because \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). Additionally, approximate \(\sin^{-1} \frac{1}{3}\). Since exact values are more complicated, utilize a calculator: \(\sin^{-1} \frac{1}{3} \approx 0.3398\).

Step 3: Calculate Inverse Cosine Values

\(\cos^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{6}\) because \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). Approximate \(\cos^{-1} \frac{1}{3}\) using a calculator: \(\cos^{-1} \frac{1}{3} \approx 1.23096\).

Step 4: Compute \(\alpha\)

Using the values from previous steps, \(\alpha = \frac{\pi}{3} + 0.3398 \approx 1.04646 + 0.3398 = 1.38626\) radians.

Step 5: Compute \(\beta\)

Using the inverse cosine values, \(\beta = \frac{\pi}{6} + 1.23096 \approx 0.5236 + 1.23096 = 1.75456\) radians.

Step 6: Compare \(\alpha\) and \(\beta\)

After calculating \(\alpha \approx 1.38626\) and \(\beta \approx 1.75456\), we compare both values. Clearly, \(\alpha < \beta\).

Step 7: Determine the Correct Answer

Since \(\alpha < \beta\), the correct option is (C) \(\alpha < \beta\).

Key Concepts

Inverse Sine

The inverse sine function, often denoted as \(\sin^{-1}(x)\), is used to find the angle whose sine value is \(x\). For example, if \(\sin \theta = x\), then \(\theta = \sin^{-1}(x)\). The range of the inverse sine function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.
This means any angle derived from the inverse sine will lie within this interval. When calculating the inverse sine of familiar values from the unit circle, such as \(\frac{\sqrt{3}}{2}\), we know that \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), so \(\sin^{-1} \left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}\).
The calculations can be approximated using technology or tables if the sine value provided does not directly correspond to a well-known angle.

Inverse Cosine

Similarly, the inverse cosine function, written as \(\cos^{-1}(x)\), finds the angle that has \(x\) as its cosine. If \(\cos \theta = x\), then \(\theta = \cos^{-1}(x)\). The range for this function is from \(0\) to \(\pi\) radians.
Understanding this range helps ensure the angle found using \(\cos^{-1}(x)\) is in the correct interval. For instance, knowing that \(\cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\), we have: \(\cos^{-1} \left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}\).
For other values, like \(\frac{1}{3}\), which are less straightforward, we often resort to a calculator for a numerical approximation, since precise exact values aren't easily derived from the standard unit circle angles.

Angle Comparison

When comparing angles calculated in radians, it is crucial to ensure all angles are measured within the same unit. After calculating \(\alpha\) and \(\beta\), both being sums of inverse trigonometric angles, comparing them determines the relationship between the two final angles.
In this particular problem, \(\alpha\) was calculated as \(1.38626\) radians and \(\beta\) was computed as \(1.75456\) radians. By subtracting these angles, one can confirm the difference and easily see which is greater.
Because \(\alpha < \beta\), it is straightforward to deduce option (C) to be correct. Recognizing the importance of direct calculation and approximation tools ensures accurate comparisons.

Radians

Radians are a crucial way of measuring angles in trigonometry. Unlike degrees, which are more commonly used in everyday contexts, radians provide a mathematical robustness ideal for calculus and higher mathematics.
One full revolution of a circle is \(2\pi\) radians, equivalent to 360 degrees. Thus, \(\frac{\pi}{3}\) radians correspond to \(60\) degrees, showing how conversions between degrees and radians can be handled. It's essential to visualize angles in terms of segments of circles to understand their radian measure.
Utilizing radians, especially in inverse trigonometry, aligns with the natural properties of trigonometric functions, simplifying their application in various equations and real-world problems.