Problem 15 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(\tan ^{-1} a+\tan ^{-1} b\), where \(a>0, b>0, a b>1\) is equal to (a) \(\tan ^{-1}\left(\frac{a+b}{1-a b}\right)\) (b) \(\tan ^{-1}\left(\frac{a+b}{1-a b}\right)-\pi\) (c) \(\pi+\tan ^{-1}\left(\frac{a+b}{1-a b}\right)\) (d) \(\pi-\tan ^{-1}\left(\frac{a+b}{1-a b}\right)\)

Step-by-Step Solution

Answer

The correct answer is (c) \(\pi + \tan^{-1}\left(\frac{a+b}{1-ab}\right)\).

Step 1: Apply the addition formula

We use the addition formula for inverse tangent, which is \(\tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right)\) when \(-1 < ab < 1\). However, we are given that \(ab > 1\). That means our angle is in the second quadrant, adding a π to our formula. Thus, the correct formula to use will be \(\tan^{-1}a + \tan^{-1}b = \pi + \tan^{-1}\left(\frac{a+b}{1-ab}\right)\) when \(ab > 1\).

Step 2: Compare the result with the options

Now we compare the result of our calculations with the options provided by the exercise. The only choice that matches our result is option (c) \(\pi + \tan^{-1}\left(\frac{a+b}{1-ab}\right)\).

Key Concepts

Addition Formula for Inverse Tangent

Understanding the addition formula for inverse tangent, also known as arctangent, is crucial for solving complex trigonometry problems. When we have two distinct angles, say \(\tan^{-1}a\) and \(\tan^{-1}b\), their sum can be represented as another inverse tangent. This specific formula is defined as:

\[\tan^{-1}a + \tan^{-1}b = \tan^{-1}\bigg(\frac{a+b}{1-ab}\bigg)\]
under the condition that \(ab\) doesn't equal 1, as this would create a division by zero situation.However, this formula directly applies only when the product \(ab\) is between -1 and 1. When \(ab > 1\), the resulting angle falls in a different quadrant, which necessitates an adjustment to the formula to reflect the correct angle measure. This condition is particularly important because it affects how angles combine on the unit circle and ultimately the correct use of the formula in applications.

Inverse Tangent Properties

The inverse tangent, or \(\tan^{-1}\), has several properties that provide a comprehensive understanding of its behavior. Firstly, the range of \(\tan^{-1}\) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This range corresponds to the angles in the first and fourth quadrants of the unit circle, where tangent values are positive and negative respectively.

Another key property is that \(\tan^{-1}\) is an odd function. This trait reflects that \(\tan^{-1}(-x) = -\tan^{-1}(x)\), preserving symmetry about the origin on a graph. The function is continuous and increasing, which means there are no discrete jumps or declines in its values over its domain. Recognizing these properties is essential when interpreting the results of trigonometric calculations and in simplifying expressions involving inverse tangent functions.

Quadrant Analysis in Trigonometry

Quadrant analysis in trigonometry is an integral part of understanding how the trigonometric functions relate to the unit circle. It enables the identification of the sign and the relative position of an angle or a point. The unit circle is divided into four quadrants:

• Quadrant I: All trigonometric functions are positive here.
• Quadrant II: Sine and cosecant are positive; cosine, secant, and tangent are negative.
• Quadrant III: Tangent and cotangent are positive; sine, cosine, and their reciprocals are negative.
• Quadrant IV: Cosine and secant are positive; sine, tangent, and their reciprocals are negative.

The addition formula for inverse tangent requires quadrant analysis as the product of \(ab\) affects where the resultant angle will lie. In the given exercise, the product \(ab > 1\) indicates that the angle falls in the second quadrant, which is why the sum of the arctangents plus \(\pi\) gives the correct answer. Being adept at quadrant analysis helps in correctly utilizing trigonometric identities and formulas, particularly when dealing with inverse functions.