If tan(π/9), x, tan(7π/18) are in arithmetic progression and tan(π/9), y, tan(5π/18) are also in ari

Q: Wait, why do we reject the negative root here? Or is there no root to reject?

In this problem, we are dealing with Arithmetic Progressions, not quadratic equations, so there are no roots to reject! The core of the problem is simply applying the definition of an A.P., which states that for three terms $a, b, c$ in A.P., $2b = a + c$. We just need to be careful with our trigonometric identities.

Q: How do I know when to use the complementary angle identity $\tan(\frac{\pi}{2} - \theta) = \cot \theta$?

Whenever you see angles that add up to $\frac{\pi}{2}$ (or $90^\circ$), that is a huge red flag to use complementary identities. In this problem, $\frac{\pi}{9} + \frac{7\pi}{18} = \frac{2\pi + 7\pi}{18} = \frac{9\pi}{18} = \frac{\pi}{2}$. Spotting these relationships is the key to unlocking the problem.

Q: Why can't we just calculate the values of $\tan(\frac{\pi}{9})$ and $\tan(\frac{7\pi}{18})$?

Because $\frac{\pi}{9}$ is $20^\circ$, which is not a standard angle like $30^\circ, 45^\circ,$ or $60^\circ$. You cannot easily find the exact value of $\tan(20^\circ)$ without complex methods. The beauty of this problem is that you don't need the exact values; the algebraic structure allows them to cancel out perfectly!

Question

If $	an(\frac{\pi}{9}), x, 	an(\frac{7\pi}{18})$ are in arithmetic progression and $	an(\frac{\pi}{9}), y, 	an(\frac{5\pi}{18})$ are also in arithmetic progression, then $|x - 2y|$ is equal to :

Options:

(A) 4

(B) 3

(C) 0

(D) 1

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem presents two sequences in Arithmetic Progression (A.P.). For the first sequence, $	an(\frac{\pi}{9})$, $x$, and $	an(\frac{7\pi}{18})$ are in A.P.

By the definition of an Arithmetic Progression, the middle term is the arithmetic mean of the surrounding terms:

$$2x = 	an\left(\frac{\pi}{9}ight) + 	an\left(\frac{7\pi}{18}ight)$$

### The Complementary Insight

Observe the relationship between the angles $\frac{\pi}{9}$ and $\frac{7\pi}{18}$. Their sum is:

$$\frac{\pi}{9} + \frac{7\pi}{18} = \frac{2\pi + 7\pi}{18} = \frac{9\pi}{18} = \frac{\pi}{2}$$

Since these are complementary angles, we apply the identity $	an(\frac{\pi}{2} - 	heta) = \cot 	heta$. Therefore, $	an(\frac{7\pi}{18}) = \cot(\frac{\pi}{9})$.

Substituting this into our equation for $x$:

$$2x = 	an\left(\frac{\pi}{9}ight) + \cot\left(\frac{\pi}{9}ight)$$

### The Power of Identities

To simplify the expression, we convert the terms into sine and cosine:

$$2x = \frac{\sin(\frac{\pi}{9})}{\cos(\frac{\pi}{9})} + \frac{\cos(\frac{\pi}{9})}{\sin(\frac{\pi}{9})}$$

Combining these over a common denominator, the numerator becomes $\sin^2(\frac{\pi}{9}) + \cos^2(\frac{\pi}{9}) = 1$. Thus:

$$2x = \frac{1}{\sin(\frac{\pi}{9}) \cos(\frac{\pi}{9})}$$

Using the double-angle identity $2 \sin 	heta \cos 	heta = \sin(2	heta)$, we multiply the numerator and denominator by $2$:

$$2x = \frac{2}{\sin(\frac{2\pi}{9})} \implies x = \frac{1}{\sin(\frac{2\pi}{9})}$$

### The Second Progression

We apply the same logic to the second A.P.: $	an(\frac{\pi}{9})$, $y$, and $	an(\frac{5\pi}{18})$. The condition for A.P. gives:

$$2y = 	an\left(\frac{\pi}{9}ight) + 	an\left(\frac{5\pi}{18}ight)$$

Note that $\frac{5\pi}{18} = \frac{\pi}{2} - \frac{2\pi}{9}$, which implies $	an(\frac{5\pi}{18}) = \cot(\frac{2\pi}{9})$. Consequently:

$$2y = 	an\left(\frac{\pi}{9}ight) + \cot\left(\frac{2\pi}{9}ight)$$

### The Final Calculation

We are tasked with finding $|x - 2y|$. Substituting our derived expressions:

$$x - 2y = \frac{1}{\sin(\frac{2\pi}{9})} - \left( 	an\left(\frac{\pi}{9}ight) + \cot\left(\frac{2\pi}{9}ight) ight)$$

Rearranging the terms to group those involving the angle $\frac{2\pi}{9}$:

$$x - 2y = \left( \frac{1}{\sin(\frac{2\pi}{9})} - \cot\left(\frac{2\pi}{9}ight) ight) - 	an\left(\frac{\pi}{9}ight)$$

The expression in the parentheses simplifies as follows:

$$\frac{1 - \cos(\frac{2\pi}{9})}{\sin(\frac{2\pi}{9})} = \frac{2 \sin^2(\frac{\pi}{9})}{2 \sin(\frac{\pi}{9}) \cos(\frac{\pi}{9})} = 	an\left(\frac{\pi}{9}ight)$$

Substituting this back, we obtain:

$$x - 2y = 	an\left(\frac{\pi}{9}ight) - 	an\left(\frac{\pi}{9}ight) = 0$$

The final result is **$0$**.

Animated Solution for Mathematics - Trigonometry: If $tan (\frac{π}{9}), x, tan (\frac{7 π}{18})$ are in arithmetic progression and $tan (\frac{π}{9}), y, tan (\frac{5 π}{18})$ are also in arithmetic progression, then $∣ x - 2 y ∣$ is equal to :

Select Answer:

Visualized Solution

First Arithmetic Progression

Complementary Angles

Substitute and Expand

Combine Fractions

Double Angle Formula

Second Arithmetic Progression

Simplify Second A.P.

Expression for $x - 2 y$

Combine Terms

Half-Angle Identities

Final Result

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Complementary Insight

The Power of Identities

The Second Progression

The Final Calculation

Similar Questions

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Select Answer:

Visualized Solution

First Arithmetic Progression

Complementary Angles

Substitute and Expand

Combine Fractions

Double Angle Formula

Second Arithmetic Progression

Simplify Second A.P.

Expression for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x−2y

Combine Terms

Half-Angle Identities

Final Result

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Complementary Insight

The Power of Identities

The Second Progression

The Final Calculation

Similar Questions

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Expression for $x - 2 y$

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