Answer
x = u cos y = 1.3 * cos (p>6)
y = u sin y = 1.3 * sin (p>6)
z = tan-1 (x/y) = tan-1 (cos y / sin y)
Work Step by Step
a) Using the Chain Rule:
Let's first express x and y as functions of u and y:
x = u cos y
y = u sin y
Now, z = tan-1 (x>y), so
dz/du = (1 / (1 + (x/y)^2)) * (cos y * dx/du + sin y * dy/du)
= (1 / (1 + (x/y)^2)) * (cos y * cos y + sin y * sin y)
= (1 / (1 + (x/y)^2)) * (cos^2 y + sin^2 y)
= (1 / (1 + (x/y)^2))
0z>0u = dz/du * du/dt = (1 / (1 + (x/y)^2)) * du/dt
Similarly,
0z>0y = dz/du * dy/dt = (1 / (1 + (x/y)^2)) * dy/dt
By expressing z directly in terms of u and y before differentiating:
z = tan-1 (x/y) = tan-1 (cos y / sin y)
0z>0u = (cos y / sin y) * (1 / (1 + (cos y / sin y)^2)) * cos y
= cos y / (sin y * (1 + (cos y / sin y)^2))
0z>0y = (cos y / sin y) * (1 / (1 + (cos y / sin y)^2)) * -sin y
= -cos y / (sin y * (1 + (cos y / sin y)^2))
b) Evaluating 0z>0u and 0z>0y at (u, y) = (1.3, p>6):
u = 1.3
y = p>6
x = u cos y = 1.3 * cos (p>6)
y = u sin y = 1.3 * sin (p>6)
z = tan-1 (x/y) = tan-1 (cos y / sin y)
0z>0u = cos y / (sin y * (1 + (cos y / sin y)^2))
= cos (p>6) / (sin (p>6) * (1 + (cos (p>6) / sin (p>6))^2))
0z>0y = -cos y / (sin y * (1 + (cos y / sin y)^2))
= -cos (p>6) / (sin (p>6) * (1 + (cos (p>6) / sin (p>6))^2))
Note: The values of 0z>0u and 0z>0y will depend on the exact value of p>6, but without a specific value, it is not possible to calculate the exact result.
