Final answer:
To convert point P₁ (2,30°,5) from spherical to Cartesian coordinates, transform using equations with given spherical values after converting angles to radians.
Step-by-step explanation:
To express the point P₁ (2,30°,5) in Cartesian coordinates, we must convert the spherical coordinates (r, θ, ϕ) into (x, y, z) coordinates. The point P₁ has a radial distance r = 2, an azimuthal angle θ = 30°, and a polar angle ϕ = 5. The Cartesian coordinates can be found using the following transformations:
- x = r ⋅ sin(ϕ) ⋅ cos(θ)
- y = r ⋅ sin(ϕ) ⋅ sin(θ)
- z = r ⋅ cos(ϕ)
Before we calculate, we must ensure that the angles are in radians. The angle 30° is equal to π/6 radians. Plugging in the given values:
- x = 2 ⋅ sin(5) ⋅ cos(π/6)
- y = 2 ⋅ sin(5) ⋅ sin(π/6)
- z = 2 ⋅ cos(5)
Calculating these values will give us the Cartesian coordinates of point P₁.