Problem 3 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the value of (i) \(\log _{10} \tan 40^{\circ}+\log _{10} \tan 41^{\circ}+\log _{10} \tan 42^{\circ}\) \(+\ldots \ldots+\log _{10} \tan 50^{\circ}\) (ii) \(\log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ}+\log _{10} \tan 3^{\circ}\) \(+\ldots \ldots \ldots . .+\log _{10} \tan 89^{\circ}\) (iii) \(\log _{3} 4 \cdot \log _{4} 5 \cdot \log _{5} 6 \cdot \log _{6} 7 \cdot \log _{7} 8 \cdot \log _{8} 9\)

Step-by-Step Solution

Answer

(i) 0, (ii) 0, (iii) 2

Step 1 - Simplifying Logarithms Sequences

For the first problem, use the properties of logarithms to simplify. According to the properties of logarithms, the addition of logarithms equals to the logarithm of the multiplication of the arguments. Hence, the sequence simplifies to \( \log _{10} (\tan 40^{\circ} \cdot \tan 41^{\circ} \cdot \ldots \cdot \tan 50^{\circ}) \). Similarly, the second problem simplifies to \( \log _{10} (\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \ldots \cdot \tan 89^{\circ}) \).

Step 2 - Calculating Result for Sequences

For the first problem, notice that the multiples of the tangents from 40 to 50 degrees cancel out to 1, thus the result is \( \log _{10} 1 = 0 \). For the second problem, similarly the multiples of the tangents from 1 to 89 degrees also cancel out to 1, so again, the result is \( \log _{10} 1 = 0 \).

Step 3 - Simplifying Logarithms Multiplication

For the third problem, the given expression is a product of logarithms, and hence can be solved by the change of base formula. After applying the change of base formula and simplifying, the expression simplifies to \( \log_{3} 9 \). By definition, this logarithm equals to \( y \) if and only if \( 3^{y} = 9 \). Hence, the solution is \( y = 2 \).

Key Concepts

Understanding the Tangent Function in Logarithmic Sequences

The tangent function, often written as \( \tan \), plays a crucial role in trigonometry. It's one of the basic trigonometric functions, representing the ratio of the opposite side to the adjacent side in a right-angled triangle. In the context of this exercise, each term inside the logarithm involves \( \tan \) of an angle.

When we encounter a sequence like \( \log_{10} \tan 40^\circ + \log_{10} \tan 41^\circ + \ldots + \log_{10} \tan 50^\circ \), the individual angles redirect us to different tangent values. The special feature here is that certain tangent values in this range complement each other:

• \( \tan(90^\circ - x^\circ) = \cot x^\circ \)
• \( \tan x^\circ \cdot \cot x^\circ = 1 \)

This complementarity often simplifies the sequence since the product of pairs of complementary angles equals 1.

The involvement of tangent in logarithmic sequences becomes especially significant, as it allows seemingly complex expressions to reduce neatly to such manageable terms.

Simplifying Sequences with Logarithmic Identities

Sequence simplification, particularly with logarithms, can majorly hinge on the power of certain identities. One primary identity used here is that the sum of logarithms is equivalent to the logarithm of the product of the arguments:

\[ \log_b M + \log_b N = \log_b (M \cdot N) \]

This identity turns seemingly long and tedious additions into one single, simple expression. For example, \( \log_{10} \tan 40^\circ + \log_{10} \tan 41^\circ + \ldots + \log_{10} \tan 50^\circ \) can be rewritten as:

• \( \log_{10} (\tan 40^\circ \cdot \tan 41^\circ \cdot \ldots \cdot \tan 50^\circ) \)

In cases like this, sequence simplification helps to notice that the product inside becomes 1 due to the specific properties of tangent and cotangent between complementary angles.

By understanding these simple yet powerful logarithmic identities, you can effortlessly reduce complex logarithmic additions to a straightforward computation like \( \log_{10} 1 = 0 \).

Exploring Logarithmic Properties

Logarithmic properties are foundational in simplifying mathematical expressions involving logs. One of the key properties used in part of this exercise is the ability to change the base of a logarithm. The change of base formula states:

\[ \log_b a = \frac{\log_c a}{\log_c b} \]

In this particular set of problems, you use properties that transform a product of logarithmic functions into something solvable. For instance, consider:

• \( \log_3 4 \cdot \log_4 5 \cdot \ldots \cdot \log_8 9 \)

Applying transformations and simplifications gives us:

• \( \log_3 9 \)

The solution to \( \log_3 9 \) is directly found by understanding the exponential relationship \( 3^y = 9 \), giving \( y = 2 \).

Such transformations and utilizations of logarithmic properties allow even complex expressions to shrink to basic calculations, emphasizing the elegance and power of logarithms.