Question

If for real values of x, cosθ = `x + 1/x`, then ______.

Options

• θ is an acute angle.

• θ is a right angle.

• θ is an obtuse angle.

• No value of θ is possible.

Answer 1

If for real values of x, cosθ = `x + 1/x`, then no value of θ is possible.

Explanation:

Given that: cosθ = `x + 1/x`

⇒ cosθ = `(x^2 + 1)/x`

⇒ x² + 1 = xcosθ

⇒ x² – xcosθ + 1 = 0

For the real value of x,

b² – 4ac ≥ 0

⇒ (–cosθ)² – 4 × 1 × 1 ≥ 0

⇒ cos²θ – 4 ≥ 0

⇒ cos²θ ≥ 4

⇒ cosθ ≥ ± 2   .......[– 1 ≤ cosθ ≤ 1]

So, the value of θ is not possible.