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Given \( x+\frac{1}{x}=3 \cos A \) and \( x^{2}+\frac{1}{x^{2}}=2 \) Determine the value of \( \cos 2 \mathrm{~A} \) without using a calculator 2

Ask by Bob Harrington. in South Africa
Mar 11,2025

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Answer

\( \cos 2A = -\frac{1}{9} \)

Solution

We start with the given equations: \[ x+\frac{1}{x}=3\cos A \quad \text{and} \quad x^2+\frac{1}{x^2}=2. \] **Step 1. Express \( x^2+\frac{1}{x^2} \) in terms of \(x+\frac{1}{x}\):** We know that \[ \left(x+\frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}. \] Thus, isolating \( x^2+\frac{1}{x^2} \) gives \[ x^2+\frac{1}{x^2} = \left(x+\frac{1}{x}\right)^2 - 2. \] **Step 2. Substitute the given expression for \( x+\frac{1}{x} \):** Since \( x+\frac{1}{x}=3\cos A \), substitute this into the previous equation: \[ x^2+\frac{1}{x^2} = (3\cos A)^2 - 2 = 9\cos^2 A - 2. \] **Step 3. Set equal to the given value of \( x^2+\frac{1}{x^2} \):** We are given that \[ x^2+\frac{1}{x^2} = 2. \] Thus, \[ 9\cos^2 A - 2 = 2. \] **Step 4. Solve for \( \cos^2 A \):** Add \(2\) to both sides: \[ 9\cos^2 A = 4. \] Then, \[ \cos^2 A = \frac{4}{9}. \] **Step 5. Use the double angle formula for cosine:** The double angle formula is given by \[ \cos 2A = 2\cos^2 A - 1. \] Substitute \(\cos^2 A =\frac{4}{9}\) into the formula: \[ \cos 2A = 2\left(\frac{4}{9}\right) - 1 = \frac{8}{9} - 1. \] **Step 6. Simplify to find \( \cos 2A \):** \[ \cos 2A = \frac{8}{9} - \frac{9}{9} = -\frac{1}{9}. \] The value of \( \cos 2A \) is therefore \[ -\frac{1}{9}. \]

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Beyond the Answer

To solve for \( \cos 2A \), we start with the equations given in the problem. Given that: \[ x + \frac{1}{x} = 3 \cos A \] we square both sides to get: \[ \left( x + \frac{1}{x} \right)^2 = (3 \cos A)^2 \] This expands to: \[ x^2 + 2 + \frac{1}{x^2} = 9 \cos^2 A \] Rearranging gives us: \[ x^2 + \frac{1}{x^2} = 9 \cos^2 A - 2 \] We also know from the problem statement that \( x^2 + \frac{1}{x^2} = 2 \). Setting this equal to our previous equation yields: \[ 2 = 9 \cos^2 A - 2 \] Adding 2 to both sides results in: \[ 4 = 9 \cos^2 A \] Solving for \( \cos^2 A \) gives: \[ \cos^2 A = \frac{4}{9} \] Next, using the double angle formula, we know: \[ \cos 2A = 2\cos^2 A - 1 \] Substituting our value of \( \cos^2 A \) into this formula: \[ \cos 2A = 2 \cdot \frac{4}{9} - 1 = \frac{8}{9} - 1 = \frac{8}{9} - \frac{9}{9} = -\frac{1}{9} \] Thus, the value of \( \cos 2A \) is: \[ \boxed{-\frac{1}{9}} \]

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