Question
A ship sailing in the Gulf Stream is heading $25.0^\circ$ west of north at a speed of 4.00 m/s relative to the water. Its velocity relative to the Earth is 4.80 m/s $5.00^\circ$ west of north. What is the velocity of the Gulf Stream? (The velocity obtained is typical for the Gulf Stream a few hundred kilometers off the east coast of the United States.)

$1.72 \textrm{ m/s, } 47.7^\circ \textrm{ E of N}$
Note: At 8:00 I should have written a negative sign in the denominator of the fraction that we're taking the inverse sign of when solving for alpha. Nevertheless, I did type the negative sign into my calculator (see the third screenshot), so the answer is correct.

# OpenStax College Physics, Chapter 3, Problem 66 (Problems & Exercises)  In order to watch this solution you need to have a subscription.

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### Calculator Screenshots

• • • Video Transcript

Hi! These videos are so great!
Question for this one is when solving b), when isolating for angle alpha by taking the inverse cosine, why isn't that equation divided by -2*Vwe*Vws ? You have 2*Vwe*Vws (aka no negative sign in front of the 2). Thanks!

Hello, thanks for the feedback and I'm really glad the videos are helpful!
It turns out I got a bit lucky here. The negative should be there, as you noticed, but it wouldn't change the answer. Cosine of a number equals the cosine of the negative of that number (see odd/even identities here: https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Reflecti…), which means the inverse cosine gives the same answer since calculators always default to the 1st quadrant (0 to 90) answer. Put in other words, looking at the cosine graph, you'll see it is symmetric about the y-axis, which is to say that each angle, and it's negative counterpart, have the same y-coordinate. This is a graphical way of saying the same thing. Inverse cosine asks the calculator "which angle has this y-coordinate on the cosine graph". Since there are an infinite number of answers (many angles have the same y-coordinate, in other words) the calculator has to pick one, and it's designed to pick the one between 0 and 90 degrees. Had I remembered the negative sign, the answer would still be the same. I'll make a note in the quick answer for other students, and thank you for noticing this.
All the best,
Shaun

Actually, I need to back-peddle that comment a bit. While it's true that cosine is symmetric about the y-axis and so on, that just means the cosine will be the same for positive angles and their respective negative counterparts. It does not mean, however, that the inverse cosine of a negative will be the same as for the inverse cosine of the respective positive. Angles (x-coordinates on the cosine graph, in other words) are different for negative vs positive y-values on the graph. In any case, I mis-identified the luck: it's actually that I input the negative sign correctly in my calculator and just forgot to write it in the video.

I didn't think to look at the calculator screenshots - that would have answered my question. Thank you so much for responding and for these helpful videos!

Hi Shaun! Thank you for the explanation! I run into a problem when I'm trying to solve "alpha" using sine-law, I got a completely different answer with yours by doing cosine-law. So I wonder why they didn't come out to be the same. The answer I got from sine law is 72.64498431 deg, and the cos-law answer is 107.2551795. When I try this,180 - 72.64 =107.3550157 deg. So I wonder if there is a trig thing going on with the first and second quadrants.