Question

Write each of the functions below in polar form: u = r cos(t − ).

a) u = r cos(t)
b) u = r sin(t)
c) u = r cos(t + )
d) u = r sin(t − )

Answer 1

The student's question pertains to expressing functions in polar form, which involves the trigonometric identities and phase shifts associated with the angle variable 't' in the context of circular motion.

Step-by-step explanation:

The question concerns the conversion of functions into polar form, which is a topic in mathematics involving trigonometric functions and their representation in terms of radius and angle, typically denoted as r and t respectively. To express the given functions in polar form:

u = r cos(t) is already in the standard polar form.

u = r sin(t) can be expressed as u = r cos(t - π/2), using the co-function identity where cos(θ) = sin(θ + π/2).

u = r cos(t + φ) is also in polar form with a phase shift of +φ.

u = r sin(t - φ) can be expressed as u = r cos(t - (φ + π/2)) in polar form.

Each function describes a particle's motion in a circular path with radius r and central angular velocity associated with t, which may involve calculations of velocity, acceleration, and centripetal force, all of which are elements of circular motion in physics.