# Thread: more precalc.. law of sines and cosines

1. ## more precalc.. law of sines and cosines

this is hard to explain without a diagram.. but ill try:
a plane is traveling 30 degrees NE at 400 mph
the wind is traveling 60 degrees NE at 34 mph
what is the new speed of the plane and its bearing (because of the effect of the wind)?
can you explain too? thanks!

2. Originally Posted by jul07010
this is hard to explain without a diagram.. but ill try:
a plane is traveling 30 degrees NE at 400 mph
the wind is traveling 60 degrees NE at 34 mph
what is the new speed of the plane and its bearing (because of the effect of the wind)?
can you explain too? thanks!
I hope this post doesn't come too late ...

1. Draw a sketch (see attachment)

2. The length of the red arrow cog (=course over ground) correspond to the actual speed of the airplane. Use the cosine rule:

$v = \sqrt{34^2+400^2-2 \cdot 34 \cdot 400 \cdot \cos(150^\circ)} \approx 429.8$

3. Calculate the angle between the blue arrow, describing the direction and the speed of the wind, and the red arrow. Use Sine rule or Cosine rule again:

$\dfrac{\sin(\alpha)}{\sin(150^\circ)}=\dfrac{400}{ 429.8}~\implies~\alpha \approx 27.732^\circ$

Thus you have to add the course of the airplane an angle of (30° - 27.732°). The airplane is flying a course over ground of N32.268°E.