I am having trouble understanding the reasoning behind the solution in this *Irodov General Physics* problem:

> **problem 1.6.)** A ship moves along the equator to the east with velocity
$v_0=30\;km/h$. The southeastern wind blows at an angle $\theta=60°$ to the equator with velocity $v=15\;km/h$. Find the wind velocity $v'$ relative to the ship and the angle $\theta'$ between the equator and the wind direction in the reference frame fixed to the ship.

My calculations:

![My diagram][1]

After drawing the initial diagram I consider that in order for the wind direction to be relative to the boat, I need to subtract the boat vector from the wind vector.

In my diagram I have repositioned the boat vector to be subtracted from the wind vector and using trigonometry I solve as follows:

First the *cosine law* to determine the magnitude

The magnitude = $\sqrt{30^2 + 15^2 - 2\cdot 15\cdot 30\cdot \cos{60}}$

= $\sqrt{675} \approx 25.98$

Then for the angle I can use *sine law*

$\frac{sin(\theta)}{30\frac {km}{h}} = \frac{sin(60)}{\sqrt{675}}$

$\theta= \arcsin{\frac{30\frac {km}{h}\cdot sin(60)}{\sqrt{675}}}$

$\theta = 90°$

The answer I get is therefore that the wind, in reference to the boat (as felt by the people standing on the boat), is approx $26\:km/h$ at an angle of $60°$ South of West (in relation to the equator).

However the answer given in the solutions is always that the approximate speed of the wind is $40\:km/h$ and the angle is approx $19°$. 

I have looked at the solutions and I think the best explanation is given in the following diagrams obtained from [link #3][6]:

![Diagram][2]

![Formula][3]

I do not understand, however, why he is using $\cos{(\pi +\theta)}$. I want to understand why my solution is incorrect and yields a different answer. What is my mistake and what is the proper intuition for understanding this problem if I have done so incorrectly?


Solutions I have viewed:

1. [This Youtube Video][4]

2. [This online blog][5]

3. [This online blog][6]


  [1]: https://i.sstatic.net/u4AlL.jpg
  [2]: https://i.sstatic.net/9rxFQ.jpg
  [3]: https://i.sstatic.net/VFCDY.jpg
  [4]: https://www.youtube.com/watch?v=lewt41RX3lQ
  [5]: http://irodovphysicssolutionrks.blogspot.in/2012/06/16.html
  [6]: http://irodovsolutionsmechanics.blogspot.in/2007/07/problem-16.html