Question

A line in the 3-dimensional space makes an angle θ `(0 < θ ≤ π/2)` with both the x and y axes. Then the set of all values of θ is the interval ______.

Options

• `(0, π/4]`

• `[π/6, π/3]`

• `[π/4, π/2]`

• `(π/3, π/2]`

Answer 1

A line in the 3-dimensional space makes an angle θ `(0 < θ ≤ π/2)` with both the x and y axes. Then the set of all values of θ is the interval `underlinebb([π/4, π/2]`.

Explanation:

It makes θ with the x and y-axis.

Let line makes α with z-axis.

`α ∈ [0, π/2]`

l = cosθ, m = cosθ, n = cos α

we have l² + m² + n² = 1

`\implies` cos²θ + cos²θ + cos² α = 1 

`\implies` 2 cos²θ = 1 – cos²α

`\implies` cos²θ = `(sin^2α)/2`

`\implies` cos θ = `sinα/sqrt(2)`

As  `α ∈ [0, π/2]` then  `θ ∈ [π/4, π/2]`