Suppose that the point (rho, theta, phi)= (12, 2pi/3, 5pi/4) in spherical coordinates can be expressed as (x, y, z) in rectangular coordinates. Find x+z. I first found x by using the conversion method (r*sin(theta)cos(phi)= x ) I got -3(sqrt(6)). Then I found z using (z= r*cos(theta) ) and got -6. I added these to get a final answer of -3(sqrt(6)) - 6 but apparently this answer is wrong.
are you sure z is not r cos phi
in math theta is usually i the xy plane
x = r sin phi cos theta
y = r sin phi sin theta
z = r cos phi
In engineering and physics these conventions are often different, nut I suspect you should be using phi as the angle down from the z axis, not theta
http://mathworld.wolfram.com/SphericalCoordinates.html
Other than that - Good job !
whats the answer?
To find the value of x and z in rectangular coordinates corresponding to the given spherical coordinates, you've correctly used the conversion formulas:
x = r * sin(theta) * cos(phi)
z = r * cos(theta)
Let's calculate these values step-by-step based on the given spherical coordinates:
r = 12
theta = 2π/3
phi = 5π/4
First, let's find x:
x = r * sin(theta) * cos(phi)
x = 12 * sin(2π/3) * cos(5π/4)
Now, to simplify this expression, let's recall some trigonometric identities. We have:
sin(2π/3) = sin(π - π/3) = sin(π/3)
cos(5π/4) = cos(π + π/4) = -cos(π/4)
Using these identities in our calculation, we get:
x = 12 * sin(π/3) * (-cos(π/4))
x = 12 * (√3/2) * (-1/√2)
x = -6 * (√3/√2)
x = -6 * (√6/2)
x = -3√6
So, the value you've found for x is correct: x = -3√6.
Now, let's find z:
z = r * cos(theta)
z = 12 * cos(2π/3)
Using the identity cos(2π/3) = -1/2, we have:
z = 12 * (-1/2)
z = -6
Now, to find x + z:
x + z = (-3√6) + (-6)
x + z = -3√6 - 6
Therefore, the final answer is indeed -3√6 - 6, which confirms that your initial calculations were correct.