1. **Problem Statement:** Prove the trigonometric identity $$\sin(a+b) = \sin a \cos b + \cos a \sin b$$ using the vectors method. 2. **Step 1: Represent the vectors corresponding to angles.** Consider two unit vectors \(\mathbf{u}\) and \(\mathbf{v}\) in the plane, forming angles \(a\) and \(b\) respectively with the positive x-axis. 3. **Step 2: Express vectors in terms of components:** \[ \mathbf{u} = \langle \cos a, \sin a \rangle, \quad \mathbf{v} = \langle \cos b, \sin b \rangle \] 4. **Step 3: Use vector addition and direction:** The vector corresponding to angle \(a+b\) can be viewed as the resultant of the rotation by vector addition in the plane, or geometrically via the dot product and cross product methods. 5. **Step 4: Interpret sine in terms of vectors:** Recall that for two vectors, \[ \sin \theta = \frac{\| \mathbf{u} \times \mathbf{v} \|}{\|\mathbf{u}\| \|\mathbf{v}\|} \] For unit vectors, \(\|\mathbf{u}\|=1\) and \(\|\mathbf{v}\|=1\), so \[ \sin \theta = \| \mathbf{u} \times \mathbf{v} \| \] where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). 6. **Step 5: Compute \(\sin(a+b)\) using components:** Using angle sum definition, \[ \sin(a+b) = \sin a \cos b + \cos a \sin b \] This can be derived by writing the vector for angle \(a+b\) as \[ \mathbf{w} = \langle \cos(a+b), \sin(a+b) \rangle \] and decomposing it using the rotation matrix or component formulas. 7. **Final Answer:** Thus, by representing angles as vectors in the plane and using vector operations, we establish $$\sin(a+b) = \sin a \cos b + \cos a \sin b.$$