A particle with a mass of 5.0 × 10^ -12 kg and a charge of 5.0 is...

Given:
Mass $m = 5.0 \times 10^{-12} \text{ kg}$
Charge $q = 5.0 \mu\text{C} = 5.0 \times 10^{-6} \text{ C}$
Speed $v = 1.0 \text{ km s}^{-1} = 1.0 \times 10^3 \text{ m s}^{-1}$
Magnetic field $B = 5.0 \text{ mT} = 5.0 \times 10^{-3} \text{ T}$
Angle $\theta = \sin^{-1}(0.90) \Rightarrow \sin\theta = 0.90$

From trigonometric identity, $\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - 0.90^2} = \sqrt{0.19} \approx 0.436$

The radius of the helical path is given by:
$r = \dfrac{mv\sin\theta}{qB}$

Substituting the values:
$r = \dfrac{5.0 \times 10^{-12} \times 1.0 \times 10^3 \times 0.90}{5.0 \times 10^{-6} \times 5.0 \times 10^{-3}}$

$r = \dfrac{4.5 \times 10^{-9}}{25.0 \times 10^{-9}} = 0.18 \text{ m} = 18 \text{ cm}$

The diameter of the helical path is:
$D = 2r = 2 \times 18 \text{ cm} = 36 \text{ cm}$

The pitch of the helical path is given by:
$p = v\cos\theta \times T = v\cos\theta \times \dfrac{2\pi m}{qB}$

Substituting the values:
$p = 1.0 \times 10^3 \times \sqrt{0.19} \times \dfrac{2\pi \times 5.0 \times 10^{-12}}{5.0 \times 10^{-6} \times 5.0 \times 10^{-3}}$

$p = 436 \times \dfrac{10\pi \times 10^{-12}}{25.0 \times 10^{-9}} = 436 \times 0.4\pi \times 10^{-3} \approx 0.548 \text{ m}$

$p \approx 54.8 \text{ cm} \approx 55 \text{ cm}$