solve the equation?

Question

a plane is heading due south with an airspeed of 282 mph. a wind from a direction of 57.0 degrees is blowing at 15.0mph. find the bearing of the plane. (note that bearings are measured from north clockwise) round results to an appropriate number of significant digits

Answer 1

Set it up as a triangle.
Side 1: 282 mph, bearing 180°
Side 2: 15 mph, bearing 57°
Side 3: s

The triangle has 3 angles:
Angle 1: 180 - 57 = 123°
Angle 2: x
Angle 3: y

Now, you can use Law of sines to find the 2nd angle (Angle x):
sin a1 / s1 = sin a2 / s2 = sin a3 / s3
sin 123 / 282.399 = sin x / 15 = sin y / s

For the portion above "sin x / 15", x is the angle opposite the 15 mph wind - i.e. the angle we're looking for.

sin x / 15 = sin 123 / 282.399
sin x = 15 * sin 123 / 282.399
sin x = -0.04455
x = 2.5532° off a bearing of 180°

Since the wind is going 57°, the bearing will be 180 + 2.5532° (it has to head into the wind), giving you a bearing of 183° (rounded to 3 sig. digits).