Chapter 8.6, Problem 28PPS

To determine

To calculate:

The angle between trail $1$ and $2$ .

Answer to Problem 28PPS

The angle between trail $1$ and trail $2$ is ${104}^{\circ }$ .

Explanation of Solution

Given information:

  

Concept used:

Law of cosines:

  $c^2=a^2+b^2-2ab\cos C$

Where,

  $a,b=$ sides

  $\cos C=$ angle

Calculation:

  

With the help of the map diagram, it is observed that having the three sides and no angles measures get the help of the law of cosines to solve the issue.

Lets $x$ be the angle between trail $1$ and trail $2$

Law of cosines:

  $c^2=a^2+b^2-2ab\cos C$

Where,

  $\begin{array}{l}a=3\\ b=2\\ c=4\end{array}$

And $\cos C=x^{\circ }$

First substitute and then simplify,

  ${\left(4\right)}^2={\left(3\right)}^2+{\left(2\right)}^2-2\left(3\right)\left(2\right)\cos \left(x\right)$

Now simplify,

  $16=9+4-12\cos \left(x\right)$

Now solve for the $\cos \left(x\right)$

  $-0.25=\cos \left(x\right)$

Now use the invers cosine ratio, the value of $x$ is,

  $x\approx 104$

Hence, the angle between trail $1$ and trail $2$ is ${104}^{\circ }$ .