= The burglar, Tom, stands on a ladder in order to be level with the castle door. Hewonders what length of rope he would need to throw a grappling hook in order to reach the largest window at the top. The castle stands at a 90 degree angle, and the height of the window is 40 feet. He knows that he would be throwing at a 50 degree angle. He uses the Law of Sines to calculate that forty feet over the sin of fifty degrees would be equal to the length of B over sin of ninety. He cross multiplies these values and then divides by sin fifty degrees. He realizes then that he will need at least 52.22 feet of rope to reach the window.
Tom realizes that he only has 50 feet of rope. Now he wonders what distance he will go by walking up the stairs. He judges that he is standing 10 feet from the castle and that he is 5 feet tall. His standing creates a 90 degree angle between his head and the bottom of the castle. He uses the Law of Cosines to calculate the distance up the stairs. He uses the equation c squared equals a squared plus b squared minus two multiplied by alpha multiplied by the distance of A multiplied by the distance of B multiplied by the cos of ninety degrees and then square roots it to find distance. He finds that the distance is 11.18 feet.
He finds the distance up the stairs far more do-able and takes that route instead in order to go into the castle. He walks up the stairs, into the castle and up the next set of stairs.
Tom then runs into the princess of the castle. He bows before her, putting on an act of respect. In his mind though, he calculates the distance from his head to the crown as he is on his knees. He knows he and the princess are standing about 3.94 feet apart. He knows that his head is 2.5 feet away from the crown. The princess standing there, staring at him, creates a 90 degree angle with his head. He wishes to know what angle her head, where the crown sits, is at. . He uses cos(alpha degrees)= 3.942.5. He then solves the equation in his head and realizes the crown sits at an angle of 50.62 degrees.
He decides that he does not wish to steal from the princess, as she offered him a drink of punch in a very nice way. They instead become friends and enjoy the party together.