For automated driving applications, exact motion planning is pivotal to obstacle-avoidance and local driving, particularly in sophisticated surroundings. Treating this problem from a differential geometric viewpoint aids in simplifying the control and planning processes. Different from the classic differential geometric methods that result in piece-wise solutions with both forward and backward moving, this paper explores the application of sinusoids in exact motion planning regarding both the positive/continuous speed requirements and the system non-holonomic constraint. By converting the vehicle kinematic model to a similar chained form, we apply sinusoidal inputs with biases to generate the reference trajectory and discuss the principles of parameter selection for a given lateral displacement. To overcome the extra constraint brought by l’Hospital rule, we introduce a pair of entry/leave transition periods to ensures an adjustable speed profile and design flexibility. The resulting trajectory reflects an strong applicability to lane-changing or evasive steering scenarios. Since the proposed method does not account for the small-time controllability because of the positive/continuous speed constraint, the reachability problem is further investigated. We provide an analytic estimation of the reachable area edge and compare it with the numerical simulation results. Finally, some indoor field-tests verified the feasibility of the proposed method.