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((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.