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Ok here is the question:a boy moves from a point X and walk 285m to Y on a bearing of 078° he then moves due south to a point Z which is 307m from X calc. The bearing of X from Z, the distance between YZ so what valuable did you declare sinz,sinx, siny so that I can substitute in

1 answer

To solve the problem step by step, let's first visualize the scenario and assign values to the points.

  1. Points and Movements:
    • Start at point X.
    • Walk from X to Y on a bearing of 078° for 285 meters.
    • From Y, walk due south to point Z for a distance of 307 meters.

Step 1: Determine the position of Y

Using the bearing to find the coordinates of point Y, we can break the movement from X to Y into its components. The bearing of 078° means it is 78° clockwise from north.

  • The coordinate system:
    • North corresponds to the positive Y-axis.
    • East corresponds to the positive X-axis.

To find the coordinates of point Y:

  • X component (Eastward) = \( 285 \cdot \sin(78°) \)
  • Y component (Northward) = \( 285 \cdot \cos(78°) \)

Step 2: Determine the coordinates of Z

Since point Z is directly south of point Y, the coordinates of Z can be determined by subtracting the distance (307 m) from the Y component of Y.

Coordinates of point Z:

  • The X coordinate stays the same as point Y.
  • Y coordinate of Z = \( Y_{y} - 307 \)

Step 3: Calculate the bearing of X from Z

To find the bearing of point X from point Z:

  1. Calculate the difference in X coordinates (Eastward) and Y coordinates (Northward) between points Z and X.
  2. Calculate the angle using the arctangent function.
  3. Convert it to a bearing (0° to 360°).

Step 4: Calculate the distance YZ

To find the distance between points Y and Z:

  1. Since we know that YZ is due south, we can just subtract the Y coordinates of Y and Z to get the distance.

Calculation of trigonometric values

For your request about declaring \( \sin z \), \( \sin x \), and \( \sin y \):

  • Let:
    • \( z \) represent the angle from Y to Z.
    • \( x \) represent the angle from Y to X.
    • \( y \) represent the angle from Z to X.

You can find the sines using the triangle formed by points X, Y, and Z:

  • \( \sin y = \frac{Y_{y} - Z_{y}}{XZ} \)
  • \( \sin x = \frac{Y_{y}}{XY} \)
  • \( \sin z = \frac{Z_{x} - Y_{x}}{YZ} \)

With this information, you can perform the calculations, substituting as needed!

Let me know if you need further assistance with any specific calculations or if you'd like to proceed with values!