To find how far Sam is from his starting point after hiking, we can use the law of cosines. The law of cosines relates the sides and angles of a triangle, which is useful when dealing with non-right triangles.

Given:

• The first leg of the hike: $a = 4.5$ km
• The second leg of the hike: $b = 3.8$ km
• The angle between the two legs: $\theta = 40^\circ$

The distance $c$ from the starting point can be found using the law of cosines:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$

Substituting the values:

$c^2 = (4.5)^2 + (3.8)^2 - 2 \cdot 4.5 \cdot 3.8 \cdot \cos(40^\circ)$

Now, let's calculate it.

$c^2 = 20.25 + 14.44 - 2 \cdot 4.5 \cdot 3.8 \cdot \cos(40^\circ)$

First, calculate $\cos(40^\circ)$:

$\cos(40^\circ) \approx 0.7660$

Then:

$c^2 = 20.25 + 14.44 - 2 \cdot 4.5 \cdot 3.8 \cdot 0.7660$

$c^2 = 20.25 + 14.44 - 26.244$

$c^2 = 8.446$

Finally, take the square root to find $c$:

$c \approx \sqrt{8.446} \approx 2.906 \text{ km}$

So, Sam is approximately 2.91 km from his starting point.

Would you like more details on the solution? Here are some related questions to expand your understanding:

1. What is the law of cosines used for in general geometry problems?
2. How would the distance change if the angle were larger or smaller?
3. Can this problem be solved using vector addition instead of the law of cosines?
4. How can we interpret the problem geometrically by drawing it out?
5. What other trigonometric identities could be used in similar problems?

Tip: Always draw a diagram when dealing with trigonometry problems involving angles and distances. It can help visualize the problem better and avoid mistakes.