How do I make my camera follow the player while also following its rotation?

My camera is able to rotate relative to the player's position but it was stay in that rotated position and not move until I press the button to make it move.. How do I make my camera be dynamic by rotating to face the player from all 4 directions while also be dynamic by following the player's movements.

Edit: I've tried using offset scripts but it doesn't and just cuases my camera to do 360s while staying behind my player

my camera script for calculating the math

 using UnityEngine;
 using System.Collections;
 
 public class CameraControll : MonoBehaviour
 {
     public Transform target;
 
     public float horizMove = 45f;
     public float VertMode = 15f;       
 
     public void MoveHorizontal(bool left)
     {
         float dir = 1;
         if (!left)
             dir *= -1;
         transform.RotateAround(target.position, Vector3.up, horizMove * dir);
 
     }
 
     public void MoveVertical(bool right)
     {       
         float dir = 1;
         if (!right)
             dir *= -1;
         transform.RotateAround(target.position, transform.TransformDirection(Vector3.right), VertMode * dir);
    }  
 }

my script for the inputs

 using UnityEngine;
 using System.Collections;
 
 public class InputManager : MonoBehaviour
 {
 
 
     CameraControll cam;
 
     void Start()
     {
         cam = GetComponent<CameraControll>();
     }
 
     // Update is called once per frame
     void Update()
     {
         if (Input.GetKeyDown(KeyCode.LeftArrow))
         {
             cam.MoveHorizontal(true);
         }
         else if(Input.GetKeyDown(KeyCode.RightArrow))
         {
             cam.MoveHorizontal(false);
         }
         else if (Input.GetKeyDown(KeyCode.UpArrow))
         {
             cam.MoveVertical(true);
         }
         else if (Input.GetKeyDown(KeyCode.UpArrow))
         {
             cam.MoveVertical(false);
         }
     }
 }