The equation y = *M* sin *Bt + N* cos *Bt* and the equation y = *A* sin *(Bt* + *C)* are equivalent where the relationships of *A, B, C, M*, and *N* are as follows. The proof is direct and follows from the sum identity for sine. The following is a summary of the properties of this relationship.

M sin Bt + N cos Bt = *(Bt* + *C)* given that Cis an angle with a point *P(M, N)* on its terminal side (see Figure 1 ).

**Figure 1**

Reference graph for *y* = *M* sin *Bt* + *N* cos *Bt*.

**Example 1**: Convert the equation *y* = *t* + 2 cos 3 *t* to the form *y* = *A* sin *(Bt + C)*. Find the period, frequency, amplitude, and phase shift (see Figure 2).

**Figure 2
**Drawing for Example 1.

**Example 2:** Convert the equation *y* = −sin π *t* + cos π *t* to the form *y* = *A* sin *(Bt* + *C)*. Find the period, frequency, amplitude, and phase shift (see Figure 3 ).

**Figure 3**

Drawing for Example 2.