Question

\(2 \tan \frac{7 \pi}{6}, 4 \tan \frac{9 \pi}{4}\) and \(8 \tan \frac{10 \pi}{3}\) are in (a) AP (b) GP (c) AGP (d) HP

Short answer

Answer: There is no progression present in these terms.

Step-by-step solution

Simplify the given expressions

We are given three expressions: 1. \(2 \tan \frac{7 \pi}{6}\) 2. \(4 \tan \frac{9 \pi}{4}\) 3. \(8 \tan \frac{10 \pi}{3}\) Let's simplify these expressions one by one using the trigonometric identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and circular functions:

Expression 1

\(2 \tan \frac{7 \pi}{6} = 2 \cdot \frac{\sin(\frac{7 \pi}{6})}{\cos(\frac{7 \pi}{6})} = 2 \cdot \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = 2 \cdot \frac{\sqrt{3}}{3}\)

Expression 2

\(4 \tan \frac{9 \pi}{4} = 4 \cdot \frac{\sin(\frac{9 \pi}{4})}{\cos(\frac{9 \pi}{4})} = 4 \cdot \frac{1}{1} = 4\)

Expression 3

\(8 \tan \frac{10 \pi}{3} = 8 \cdot \frac{\sin(\frac{10 \pi}{3})}{\cos(\frac{10 \pi}{3})} = 8 \cdot \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = 8 \cdot -\frac{\sqrt{3}}{3}\)

Check for Arithmetic Progression (AP)

For three terms to be in AP, the middle term must be the arithmetic mean of the first and third terms: \(4 = \frac{2 \cdot \frac{\sqrt{3}}{3} + 8 \cdot -\frac{\sqrt{3}}{3}}{2}\) \(4 = \frac{-2 \cdot \frac{\sqrt{3}}{3}}{2}\) \(4 \neq -\sqrt{3}\) The terms are not in Arithmetic Progression, so option (a) can be eliminated.

Check for Geometric Progression (GP)

For three terms to be in GP, the middle term must be the geometric mean of the first and third terms: \(4 = \sqrt{2 \cdot \frac{\sqrt{3}}{3} \cdot 8 \cdot -\frac{\sqrt{3}}{3}}\) \(4 = \sqrt{-32}\) Since we cannot take the square root of a negative value, the terms cannot be in Geometric Progression. So, option (b) can be eliminated.

Check for Arithmetic-Geometric Progression (AGP)

For three terms to be in AGP, the middle term must be the square root of the product of the terms on either side: \(4 = \sqrt{(2 \cdot \frac{\sqrt{3}}{3}) \cdot (8 \cdot -\frac{\sqrt{3}}{3})}\) \(4 = \sqrt{-32}\) Again, since we cannot take the square root of a negative value, the terms cannot be in Arithmetic-Geometric Progression. So, option (c) can be eliminated.

Check for Harmonic Progression (HP)

For three terms to be in HP, the reciprocals of the terms must be in Arithmetic Progression: Let's take the reciprocals of the terms: 1. \(\frac{1}{2 \cdot \frac{\sqrt{3}}{3}} = \frac{\sqrt{3}}{2}\) 2. \(\frac{1}{4}\) 3. \(\frac{1}{8 \cdot -\frac{\sqrt{3}}{3}} = -\frac{\sqrt{3}}{8}\) Now, check if these reciprocals are in Arithmetic Progression: \(\frac{1}{4} = \frac{\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{8}}{2}\) \(\frac{1}{4} = \frac{\frac{3\sqrt{3}}{8}}{2}\) \(\frac{1}{4} \neq \frac{3\sqrt{3}}{16}\) The reciprocals are not in Arithmetic Progression, so the terms cannot be in Harmonic Progression. So, option (d) can be eliminated. Since none of the options can be confirmed, there is no progression present in these terms.

Key concepts

Arithmetic Progression (AP)

An arithmetic progression, or AP, is a sequence where each term after the first is the sum of the previous term plus a constant. This constant is called the common difference and is denoted by the letter \(d\). For a sequence \(a, a+d, a+2d, \ldots \), each term can be expressed as:

• The formula for the \(n\)-th term: \(a_n = a + (n-1)d\)
• The sum of the first \(n\) terms: \(S_n = \frac{n}{2} \times (2a + (n-1)d)\)

In this exercise, we checked if the given trigonometric expressions could form an arithmetic progression. The criteria for three numbers forming an AP is if the middle term is equal to the arithmetic mean of the first and last terms. However, from our calculations, the middle term was not equal to the mean, eliminating AP as a possibility.

Geometric Progression (GP)

Geometric progression, or GP, is a sequence where each subsequent term is found by multiplying the previous term by a fixed number, known as the common ratio, represented by \(r\). For a sequence \(a, ar, ar^2, \ldots \), this common ratio can be derived if we know any two consecutive terms using the formula \( r = \frac{a_2}{a_1} \).

• The formula for the \(n\)-th term: \(a_n = ar^{(n-1)}\)
• The sum of the first \(n\) terms: \(S_n = a \frac{r^n - 1}{r - 1}\), if \(r eq 1\)

In our problem, the test for GP involved checking if the square of the middle term equaled the product of the other two. However, we found a negative value inside the square root function, which means these terms do not form a geometric progression. Thus, GP was ruled out as well.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domain. Common trigonometric functions include sine, cosine, and tangent, with identities like:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Tangent and Cotangent Identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \)

In the exercise, simplifying expressions involved using the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), which is crucial for converting complex trigonometric terms into simpler fractions or evaluating their signs and values at specific angles. By doing this, we clarified the numerical values of our expressions to determine the nature of their progression relationship.

Harmonic Progression (HP)

A harmonic progression is unique in that it is not directly a stand-alone series but is based on the reciprocals of an arithmetic progression. Simply put, for a sequence to be in HP, the inverse of its terms should form an arithmetic progression. If you have a sequence \( a, b, c \), in HP, the sequence \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) should form an AP.

• The terms will look like: \( \frac{1}{a_1}, \frac{1}{a_1 + d}, \frac{1}{a_1 + 2d}, \ldots \)

In our situation, we took the reciprocals of the given expressions to check for this type of sequence, yet found no consistent difference indicating an arithmetic progression among those reciprocals. Therefore, the exploration confirmed that these terms could not form a harmonic progression.