Well, what you describe is exactly what you do there. Sin() and Cos() returns a value between -1 and 1. You multiply this value by your radius so the value is in between -radius and radius. Then you use it as absolute position. This way you will always rotate around 0,0,0

What you need is an offset. This can be saved at start like this:

using UnityEngine;
using System.Collections;

public class CircularMotionScript : MonoBehaviour
{
    public float mPosition = 0F;
    public float mRadius = 2.5F;
    private Vector3 m_InitialPosition;

    void Start()
    {
        m_InitialPosition = transform.position;
    }

    void Update ()
    {
        mPosition += Time.deltaTime * 1.0f; // 1.0f is the rotation speed in radians per sec.
        Vector3 pos = new Vector3(Mathf.Sin(mPosition), Mathf.Cos(mPosition), 0);
        transform.position = m_InitialPosition + pos * mRadius;
    }
}

You don't need to do all of that.. just make the object you want to move circularly a child of a stick (cube game object) , put your object at the tip of the stick... then attach a script to the stick to make it rotate function Update(){ transform.Rotate(0,0,Time.deltaTime); } make sure not to render the stick in camera..

the stick length is the radius of the circle.......