Do you use the inner or outer degrees of a right-angled triangle when calculating the vertical component of a vector

Question

I'm given the assignment of finding the vertical component of vector a - b. Below is an image of vector A. ||A|| = 6. I need to calculate side y.

I was following the assignment and tried solving y by doing; y = 6sin(50) since 360-310=50. When I checked the results it turned out I was wrong because according to khanacademy you must calculate y by 6sin(310).

So my question: Why must you use the 310 degrees rather than 50 degrees? I thought you always needed to use the amount of degrees of the inside of a right-angled triangle...

Answer 1

Vectors have direction and magnitude, but they don't have a fixed "initial point". All vectors pointing in the same direction and of the same length are equivalent.

So, translate the vector so that its base point is the origin. It will then be pointing in the same direction as it is now--sort of southeast-ish. Now measure the angle that this vector has with the positive $x$-axis (always measuring counterclockwise!!!). It is $310^\circ$, or $-50^\circ$.

The fact that this translated vector points from the origin into the 4th quadrant should tell you that its horizontal component ($x$-coordinate of the terminal end) is positive and that its vertical component ($y$-coordinate of the terminal end) is negative.

Answer 2

Because $\sin(50\,^{\circ})$ is positive, which is incorrect for your negative $y$-value. Using 310 will give you the correct sign for $y$.