91Ó°ÊÓ

The law of cosine mainly relates the triangle’s lengths of its sides to the cosine angle of the triangle. It is used for identifying the angle of the triangle.

The given data can be listed below as:

• Ricardo walks to a distance of $26.0m$to$60.0°$west of north.
• Jane walks to a distance of $16.0m$to $30.0°$south of west.

The free body diagram that shows the distance between Jane and Richard and the distance covered by them and the angle between them has been provided below,

a)

From the law of cosines, the distance between Ricardo and Jane is expressed as:

$C=\sqrt{a^2+b^2-2ab\cos y}$

Here, c is the distance between Ricardo and Jane, a and b are the distance covered by Jane and Richard, which are $26.0mand16.0m$respectively, and $\mathrm{γ}$is the angle between the persons, which is $60°$.

Substituting the values in the above expression, we get,

$$\begin{gathered} c=\sqrt{{\left(26.0m\right)}^2+{\left(16.0m\right)}^2-2.26m.16m.\cos 60°} \\ =\sqrt{676m^2+256m^2-416m^2} \\ ≈22.71m \end{gathered}$$

Thus, the distance between Ricardo and Jane is 22.71m .

b)

From the law of sine, the direction in which Ricardo should walk is expressed as:

$\frac{\sin r}{b}=\frac{\sin 60°}{c}$

Here $r$is the angle of the direction in which Ricardo should walk in the southeast direction.

Substituting the values in the above equation, we get,

$$\begin{gathered} \frac{\sin r}{16.0m}=\frac{0.866}{22.71m} \\ \sin r=0.610 \\ r={\sin }^{-1}\left(0.61\right) \\ r≈37.58° \end{gathered}$$

Thus, the direction in which Ricardo should walk is$37.58°$in the southeast direction.