If for real x, `"cos"theta=x+(1)/(x)`, then

Options

A. `theta` is an acute angle
B. `theta` is a right angle
C. `theta` is an obtuse angle
D. no value of `theta` is possible

Answer

To solve the problem, we start with the equation given:

\[ \cos \theta = x + \frac{1}{x} \]

### Step 1: Rearranging the Equation
First, we can rewrite the equation in a more manageable form:

\[ \cos \theta = x + \frac{1}{x} \]

### Step 2: Finding a Common Denominator
To combine the terms on the right side, we find a common denominator:

\[ \cos \theta = \frac{x^2 + 1}{x} \]

### Step 3: Cross Multiplying
Next, we cross-multiply to eliminate the fraction:

\[ x \cos \theta = x^2 + 1 \]

### Step 4: Rearranging into a Quadratic Equation
Now, we rearrange this equation into a standard quadratic form:

\[ x^2 - x \cos \theta + 1 = 0 \]

### Step 5: Identifying Coefficients
We can identify the coefficients of the quadratic equation \(ax^2 + bx + c = 0\):

- \(a = 1\)
- \(b = -\cos \theta\)
- \(c = 1\)

### Step 6: Applying the Discriminant Condition
For \(x\) to be a real number, the discriminant of the quadratic equation must be non-negative:

\[ D = b^2 - 4ac \geq 0 \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ (-\cos \theta)^2 - 4(1)(1) \geq 0 \]

This simplifies to:

\[ \cos^2 \theta - 4 \geq 0 \]

### Step 7: Rearranging the Inequality
Rearranging gives us:

\[ \cos^2 \theta \geq 4 \]

### Step 8: Analyzing the Result
Taking the square root of both sides, we find:

\[ |\cos \theta| \geq 2 \]

### Step 9: Conclusion
However, the range of \(\cos \theta\) is limited to \([-1, 1]\). Therefore, it is impossible for \(|\cos \theta|\) to be greater than or equal to 2.

Thus, we conclude that:

\[ \text{No value of } \theta \text{ is possible.} \]

The correct answer is option D.

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