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Answer:
- CW from north: -78.58 mph
- CCW from +x: -112.2 mph
Explanation:
A couple of different references are used when talking about angles in the x-y coordinate plane. Conventionally, angles are measured CCW from the +x axis of the plane. This is the reference usually used in physics problems involving acceleration, velocity, and displacement.
If the angles refer to bearings on a map, the usual mapping is north = +y, east = +x, and angles are measured CW from north. This is the reference usually used in (flat earth) navigation problems involving airspeed and ground speed.
We cannot tell the intended meaning of "direction angle" here, hence there are two answers. Take your choice.
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If the angle is measured counterclockwise from the +x direction, the (x, y) coordinates of the vector (137; 235°) are ...
(x, y) = 137(cos(235°), sin(235°))
(x, y) = (-78.58, -112.22)
The y-component is -112.22 miles per hour.
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If the "direction angle" is a bearing measured clockwise from the +y direction, then the x- and y-coordinates are swapped from those listed above.
(x, y) = (-112.22, -78.58)
The y-component is -78.58 miles per hour.
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Additional comment
We say this is "flat earth" navigation because any serious navigation calculation must take into account the Earth's curvature. A path that follows a great circle will not have a constant compass bearing except along the equator or a longitude line.