Final answer:
The sternothyroid muscle is the strap muscle adjacent to the trachea and posterior to the thyroid and common carotid arteries, involved in controlling the position of the larynx's thyroid cartilage.
Step-by-step explanation:
The strap muscle that is adjacent to the trachea and is posterior to the thyroid and the common carotid arteries is the sternothyroid muscle. This muscle plays a crucial role in controlling the position of the larynx's thyroid cartilage, which is important for producing different tones of voice. The sternothyroid muscle, along with other infrahyoid muscles like the omohyoid and sternohyoid muscles, generally depress the hyoid bone and are involved in raising and pulling the hyoid bone anteriorly.