Consider this: "A boat sails $6$ km West, then $5$ km Northwest. Use trigonometry to find the boat's distance and bearing from its starting point."
To solve this, I first drew on paper a shape sort of like this:
I used a protractor and ruler to make sure the lines were to scale and with accurate angles/bearing.
What I did from there was take $\tan^{-1}\frac{5}{6}$ to find the angle of $O$, which was approximately $39.8^{\circ}$. Then I did this calculation to find the length of $OB$:$$\sin 39.8=\frac{5}{x}$$
$$x\sin39.8=5$$
$$x=\frac{5}{\sin 39.8}$$
$$x=7.8$$
So there I have my answer. The distance from the starting point is $7.8$ km and the bearing relative to the starting point is $39.8^{\circ}$. However, when I use my ruler and protractor to get an approximate on paper, I get a wildly different result. I made $1$ cm equal $1$ km on my drawing, so the line $OB$ should be $7.8$ centimeters, but it's a lot closer to $10$ centimetres. And $\angle O$ is approximately $21^{\circ}$ when I use my protractor to verify.
So, did I do something wrong? I know paper will always be inaccurate compared to exact calculations, but the difference seems way too big. Did I go wrong somewhere?
Best Answer
Your calculations are faulty. First the answer:
By the law of cosines $$OB^2 = 6^2+5^2 - 2 \cdot 6\cdot 5 \cos(135^\circ)$$ gives $OB \approx 10.2$.
Now to your mistake:
Note that the coordinates of $B$ is not $[-6,5]$ but $[-(6+5/\sqrt{2}), 5/\sqrt{2}]$
Your method is fine.
Note: Added as an after thought.
In your calculations, you mistakenly used $5$Km North, instead of $5$Km Northwest. I see where your mistake is.