The diagram shows a vector 𝐀 that has a horizontal component with a magnitude of 39. The angle between the vector and the 𝑥-axis is 18 degrees. What is the magnitude of the vector? Give your answer to the nearest integer.
By the magnitude of vector 𝐀, what we mean is the size of this line. The information we’re given is that the horizontal component has a magnitude or size of 39 and that the vector makes an angle of 18 degrees with the horizontal or 𝑥-axis. We can solve this by drawing a right-angled triangle, where we know the size of one side, which is 39, and this angle 𝜃, which is 18 degrees. And the value we’re trying to find is the length of this side, which we’ll call 𝐴.
Now, relative to the angle 𝜃, the side with the magnitude we know of 39 is the adjacent. And the unknown side we’ve labeled 𝐴 is the hypotenuse. Now, recall from SOHCAHTOA that if we have the adjacent and we want to find the hypotenuse, we need to use the cosine of the angle. And this tells us that the cos of the angle 𝜃 is equal to the adjacent divided by the hypotenuse.
Now, if we put our values into this equation, we have the cos of the angle of 18 degrees, and that is equal to 39 divided by 𝐴. Now, we’re going to multiply both sides by 𝐴 and divide both sides by the cos of 18 degrees. And we find that 𝐴 is equal to 39 divided by the cos of 18 degrees. Now, we need to make sure that our calculator is in degrees, and we’ll find that 𝐴 is equal to 41.007. Now, the question asks for the value to the nearest integer. So, the answer becomes that the magnitude of vector 𝐀 is equal to 41.