Trigonometric formula of 3D mathematics

Triangle has two expressions , One is the angle that humans like Degree, One is the curve that mathematics likes Redian.

The conversion between them is
$$\begin{gathered} red=deg\ast (\pi /180)=deg\ast 0.01745329 \\ deg=red\ast (180/\pi )=red\ast 57.29578 \end{gathered}$$
We don't have to write , The general mathematical library contains the mutual conversion of angles and radians

Mathf.Red2Deg : 0.01745329

Mathf.Deg2Red ：57.29578

sine sin and cosine cos, There are other forms of trigonometric functions
$$\begin{gathered} \sec \theta =\frac{1}{\cos \theta }\csc \theta =\frac{1}{\sin \theta } \\ \tan \theta =\frac{\sin \theta }{\cos \theta }\cot \theta =\frac{1}{\tan \theta }=\frac{\cos \theta }{\sin \theta } \end{gathered}$$
Trigonometric identity
$${\sin }^2\theta +{\cos }^2\theta =1$$

General math libraries also provide inverse trigonometric functions , Such as Asin ,Acos, Atan, They can convert it into angles

Trigonometric functions are represented in the unit circle
$$\begin{gathered} x=\cos \theta \\ y=\sin \theta \\ \sin {\theta }^2+\cos {\theta }^2=1 \end{gathered}$$

Other formulas in triangles , Such as sine theorem , Cosine theorem

Sine theorem
$$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$
Cosine theorem
$$\begin{gathered} a^2=b^2+c^2-2bc\cos A \\ {b}^2=a^2+c^2-2ac\cos B \\ {c}^2=a^2+b^2-2ab\cos C \end{gathered}$$

Here is a demonstration of the random rotation of an object , Identify an object that rotates around , Then generate multiple objects to rotate around it .

Center of rotation

    public class TriangleText : MonoBehaviour
    {
    
        [SerializeField] private int _count = 5;
        [SerializeField] private float _radius = 2f;
        [SerializeField] private GameObject _prefab;
        void Start()
        {
    
            // Obtain radians 
            float red = 2 * Mathf.PI / _count;
            for (int i = 0; i < _count; i++)
            {
    
                // Get the location 
                Vector3 position = new Vector3(
                    Mathf.Cos(red * i) * _radius, transform.position.y, Mathf.Sin(red * i) * _radius);
                // Obtain direction 
                Quaternion quaternion = Quaternion.FromToRotation(Vector3.up, (transform.position - position).normalized);
                // establish 
                Instantiate(_prefab, position + transform.position, quaternion);
            }
        }
    }

A central point

Rotating objects

    [SerializeField] private float _rotationSpeed = 30f;
    [SerializeField] private Transform _point;
    private float _distance;
    private float _currentRed;
    private Quaternion _rondomRoation;

    private void Start()
    {
    
        _rondomRoation = Quaternion.AngleAxis(Random.Range(0f, 180f), Random.onUnitSphere);
        _rondomRoation = _rondomRoation.normalized;
        Debug.Log(_rondomRoation);

        _distance = (_point.position - transform.position).magnitude;

        _currentRed = Vector3.Angle(Vector3.right, transform.position) * Mathf.Deg2Rad;
        if (Vector3.Cross(Vector3.right, transform.position).y < 0)
            _currentRed += Mathf.PI;
        
        StartCoroutine(Rotation());
    }

    private IEnumerator Rotation()
    {
    
        while (true)
        {
    
            _currentRed = (_currentRed + _rotationSpeed * Mathf.Deg2Rad * Time.deltaTime) % 360f;
            float x1 = Mathf.Cos(_currentRed) * _distance;
            float y1 = Mathf.Sin(_currentRed) * _distance;
            
            Vector3 position = new Vector3(x1, 0, y1);
            position = _rondomRoation * position + _point.position;
            
            Quaternion quaternion =
                Quaternion.FromToRotation(Vector3.up, (_point.position - transform.position).normalized);
            
            transform.SetPositionAndRotation(position, quaternion);
            yield return null;
        }
        // ReSharper disable once IteratorNeverReturns
    }

object

design sketch