Problem 19 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The value of \(\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)\) is (a) 1 (b) \(\frac{1}{\sqrt{7}}\) (c) \(\sqrt{7}\) (d) None.

Step-by-Step Solution

Answer

The value of the expression \(\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)\) is equal to 1. Hence the correct answer is (a) 1.

Step 1: Simplify the Expressions

The expression we received is \(\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)\). The angle in each expression are fractions of \(\pi\) which can be simplified.

Step 2: Use the Trigonometric Identity

The trigonometric identity for multiples of tangent is \(\tan{A}\tan{B}\tan{C} = \tan{A} + \tan{B} + \tan{C}\) when \(A+B+C = \pi\). Here, \(\frac{\pi}{7} + \frac{2\pi}{7} + \frac{3\pi}{7} = \pi\), hence the identity can be used on these three expressions.

Step 3: Conclusion

After applying the trigonometric identity, the expression simplifies to \(\tan \left(\frac{\pi}{7}\right) + \tan \left(\frac{2 \pi}{7}\right) + \tan \left(\frac{3 \pi}{7}\right)\). This gives a sum of 1, therefore the answer should be (a) 1.

Key Concepts

Tangent of Angles

The tangent function is one of the primary trigonometric functions that relate an angle in a right triangle to the ratio of the side opposite the angle to the side adjacent. In this exercise, we are dealing with the tangent of specific angles: \( \frac{\pi}{7} \), \( \frac{2\pi}{7} \), and \( \frac{3\pi}{7} \). These angles are expressed in radians, which is another way of measuring angles aside from degrees. Radians make it easier to express angles in terms of \(\pi\), a constant that naturally appears in trigonometric calculations. To better understand this, think of the unit circle in trigonometry where the tangent of an angle \(\theta\) can be defined as the \(y\) coordinate of the point on the unit circle divided by the \(x\) coordinate. The tangent values will repeat in a pattern as the angle increases, leading us to various useful trigonometric identities. But the core idea remains the same: relate an angle to a ratio, making it a vital element in solving more complex trigonometric identities and equations.

Angle Sum Identities

Angle sum identities are crucial in trigonometry. They simplify trigonometric expressions and evaluate complicated function values. For tangent, the identity looks a bit different than sine and cosine. It involves combining three tangent terms when their angles sum up to \(\pi\). The trigonometric identity used here states that when you have multiples of tangent such as \(\tan{A}\tan{B}\tan{C}\) and the condition \(A+B+C = \pi\) is met, the expression simplifies to \(\tan{A} + \tan{B} + \tan{C}\).
This identity comes in handy often because it reduces complex multiplications of tangent values into a simpler addition. In our problem, the sum of the angles \(\frac{\pi}{7} + \frac{2\pi}{7} + \frac{3\pi}{7}\) equals \(\pi\), meeting the necessary condition to apply this identity.

• Start by confirming the sum of the angles equals \(\pi\).
• Apply the identity to transform the product into a simpler sum.
• Perform the addition as instructed by the identity.

This technique not only simplifies expressions but also aids in solving various trigonometric problems efficiently.

Trigonometric Simplification

Trigonometric simplification involves reducing a complex trigonometric expression into a simpler form. This often employs identities, strategic substitutions, and algebraic manipulations. In our exercise, after applying the angle sum identities, we simplified the original expression \(\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)\) to a simpler addition of tangent values.
The goal of simplification is:

• To make an expression easier to work with.
• To find exact values quickly, such as verifying options in a multiple-choice question.

In this instance, knowing the correct trigonometric identity allowed us to leapfrog through various unnecessary calculations, reaching the simplified sum. The problem's conclusion showed that this sum equaled 1, thus identifying the correct multiple-choice answer to the problem. Simplifying trigonometric expressions speeds up calculations and clarifies results, demonstrating the powerful utility of mastering identities and simplification methods.