Given are two angles start and end within the range 0..360, now rotate a point around a circle from start to end taking either the clockwise or counter-clockwise direction.

CircleMath.cs

using System.Collections;
using UnityEditor;
using UnityEngine;

public class CircleMath : MonoBehaviour
{
	public Vector3 position = Vector3.right;
	public float angleStart = 20;
	public float angleEnd = 270;
	public float duration = 1f;

	private bool animating;

	private void OnValidate()
	{
		angleStart = Mathf.Repeat(angleStart, 360f);
		angleEnd = Mathf.Repeat(angleEnd, 360f);
	}

	//[EditorButton("Counter Clockwise", Scope.PlayMode, 1f)]
	//[EditorButton("Clockwise", Scope.PlayMode, -1f)]
	private void DoAnimation(float direction)
	{
		float start = angleStart;
		float end = angleEnd;

		if (direction < 0f)
			end = WrapAround(angleEnd);
		else
			start = WrapAround(angleStart);

		StopAllCoroutines();
		StartCoroutine(TurnImpl(start, end));
	}

	private static float WrapAround(float angle)
	{
		return angle > 180f ? angle - 360f : angle;
	}

	private IEnumerator TurnImpl(float start, float end)
	{
		animating = true;

		for (float elapsed = 0; elapsed < duration; elapsed += Time.deltaTime)
		{
			float t = elapsed / duration;
			float angle = Mathf.Lerp(start, end, t);
			position = PositionFromAngle(angle);
			yield return null;
		}

		yield return null;
		animating = false;
	}

	private static Vector3 PositionFromAngle(float angle)
	{
		var x = Mathf.Cos(angle * Mathf.Deg2Rad);
		var y = Mathf.Sin(angle * Mathf.Deg2Rad);
		return new Vector3(x, y, 0f);
	}

	private void OnDrawGizmos()
	{
		Gizmos.color = Color.white;
		Gizmos.DrawWireSphere(transform.position, 1);

		Gizmos.color = Color.green;
		var pos = PositionFromAngle(angleStart);
		Gizmos.DrawSphere(pos, 0.05f);
		pos += Vector3.right * 0.1f;
		Handles.Label(pos, "Start");

		Gizmos.color = Color.yellow;
		pos = PositionFromAngle(angleEnd);
		Gizmos.DrawSphere(pos, 0.05f);
		pos += Vector3.right * 0.1f;
		Handles.Label(pos, "End");

		if (animating)
		{
			Gizmos.color = Color.red;
			Gizmos.DrawSphere(position, 0.1f);
		}
	}
}