Question

find the distance between (4, 200°) and (2, 140°) to the nearest tenth. help

Answer 1

Notice that the given points are in polar coordinates.

Recall that the distance between two points (r₁,θ₁) and (r₂,θ₂) in polar coordinates is given by the distance formula:

$d=\sqrt[]{r^2_1+r^2_2-2r_1r_2\cos (\theta_2-\theta_1)}$

Substitute (r₁,θ₁)=(4,200º) and (r₂,θ₂)=(2,140º) into the formula:

$d=\sqrt[]{4^2+2^2-2\cdot4\cdot2\cos (140-200)}$

Simplify the expression on the right:

$d=\sqrt[]{16+4-16\cos(-60)}=\sqrt[]{20-16(\frac{1}{2})}=\sqrt[]{20-8}=\sqrt[]{12}=2\sqrt[]{3}$

Express the number as a decimal to the nearest tenth as required:

$2\sqrt[]{3}\approx3.5$

Hence, the distance between the points is about 3.5 units.