[Math] Bearing and Course Trigonometry Question

trigonometry

This is the question:

A jet flew 140 mi on a course of 196 degrees and then 120 mi on a course of 106 degrees then the jet return to its starting point via the shortest route possible. Find the total distance that the jet traveled.

I found the answer for this. It's this one:
http://ph.answers.yahoo.com/question/index?qid=20110828043233AAVgQcy

Now I understand everything else, but my problem is why and how they used the given angles to confirm that it is a right triangle. I know this has something to do with theorems on parallel lines, but I just can't picture how. I will need an illustration to understand this. Thank you. 🙂

Best Answer

It's always good to draw a picture in these cases. A right angle is 90 degrees. So if you're traveling at 196 degrees, and you change to 106 degrees, you have made a change of 90 degrees; therefore you have turned at a right angle.

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[Math] Trigonometry Airplane question. Finding bearing and distance.

Cosine Law would be easier, but let's do it your way. Consider the small right triangle (call it $\Delta ABC$), which roughly looks like:

A----B
|   /
|  /
| /
|/
C

Note that $\angle ABC =59^\circ$. Thus, $AB=640\cos(59^\circ)$ and $AC=640\sin(59^\circ)$.

Now consider the large right triangle. Its two legs are $AC$ and $AB+480$. Hence, by the Pythagorean Theorem, we have: $$x=\sqrt{(AC)^2 + (AB+480)^2}=\sqrt{(640\cos(59^\circ)+480)^2 + (640\sin(59^\circ))^2} \approx 978 \text{ miles}$$ Finally, for the angle bearing: $$\theta=\tan^{-1}\left(\frac{AC}{AB+480}\right) = \tan^{-1}\left(\frac{640\sin(59^\circ)}{640\cos(59^\circ)+480}\right) \approx 34^\circ \text{ to get the bearing you subtract that value from 90 to get } S56^\circ W$$

[Math] Calculate depth using triginometry

I didn't recheck your calculations so I'll assume your approximations are correct. Other than that you did a fine job finding the depth. This problems basically boils down to finding the altitude of a triangle given two angle measures. Your approach is right on the money and exactly the approach I would use had I remembered the law of sines lol.

Best Answer

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[Math] Trigonometry Airplane question. Finding bearing and distance.

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