A pilot is flying a Cessna airplane at 180 mph (airspeed).He would like to fly in the direction N45W, but there is a 32 mph wind in the S60W direction.What direction should the pilot set his course for in order to fly along his desired track? What will his speed relative to the ground be?

To determine the direction the pilot should set his course for, you'll need to consider the vector addition of the airplane's airspeed (180 mph) and the wind's speed (32 mph). This can be done by breaking down the velocities into their horizontal and vertical components.

The given direction is N45W, which means the pilot wishes to fly 45 degrees west of north. To find the vertical and horizontal components of this desired direction, you can use trigonometry.

- The vertical component (north-south) can be found using sin(theta) = vertical component / hypotenuse, where theta is the angle from the x-axis (west) to the hypotenuse (desired direction).
- The horizontal component (east-west) can be found using cos(theta) = horizontal component / hypotenuse.

Using the given values, we can find the vertical and horizontal components for the desired direction:

- Vertical component (north-south):
sin(45) = vertical component / 1
vertical component = sin(45) = 0.707

- Horizontal component (east-west):
cos(45) = horizontal component / 1
horizontal component = cos(45) = 0.707

Now let's consider the wind's velocity. The wind is blowing in the S60W direction, meaning it is 60 degrees west of south. To break down the wind's velocity into horizontal and vertical components, you can follow the same process:

- Vertical component (north-south):
sin(60) = vertical component / 1
vertical component = sin(60) = 0.866

- Horizontal component (east-west):
cos(60) = horizontal component / 1
horizontal component = cos(60) = 0.5

To determine the resultant velocity, you need to add the corresponding components:

- Vertical component:
0.707 (airplane's north-south component) + (-0.866) (wind's north-south component) = -0.159

- Horizontal component:
0.707 (airplane's east-west component) + (-0.5) (wind's east-west component) = 0.207

Now, you have the vertical and horizontal components of the resultant velocity. To find the speed and direction of the groundspeed, you can use the Pythagorean theorem and inverse trigonometric functions.

- Speed (resultant velocity):
Speed = sqrt(vertical component^2 + horizontal component^2) = sqrt((-0.159)^2 + (0.207)^2) ≈ 0.26

- Direction (resultant velocity):
tan(theta) = vertical component / horizontal component
theta = tan^(-1)(-0.159 / 0.207) ≈ -36.2 degrees (south of west)

Therefore, the pilot should set his course at N45W (45 degrees west of north) with a speed of approximately 0.26 mph and in a direction approximately 36.2 degrees south of west relative to the ground.

we want 180(cos135,sin135) = 180(cosθ,sinθ) + 32(cos210,sin210)