Solving Arctan(x/y) - 90degrees

[flexifirm]

This is fun...

prove that:

arctan(x/y) - 90degrees = -arctan(y/x)

I had to do this recently for an electrical engineering problem (in an interview actually). It's ahrd at first, but once you picture things it comes fast.

 

[cookiemonster]

arctan(x/y) is the angle of a right triangle opposite x, so subtract 90 degrees and you're going to get the negative (since the first angle is less than 90) of the angle opposite side y.

Did you have to do that on the spot in your head? Sounds like a good question.

cookiemonster

 

[tacman]

First, let's rewrite the equation using the inverse tangent identity: arctan(x/y) = arctan(y/x) - 90degrees. This is because the inverse tangent of x/y is equal to the inverse tangent of y/x subtracted by 90 degrees.

Next, we can use the fact that the tangent function is an odd function, meaning that tan(-x) = -tan(x). This allows us to rewrite the equation as -arctan(x/y) = -arctan(y/x) + 90degrees.

Now, we can simplify the equation by multiplying both sides by -1, giving us arctan(x/y) = arctan(y/x) - 90degrees. This is the same equation we started with, but now we can see that both sides are equal.

Therefore, we have proven that arctan(x/y) - 90degrees = -arctan(y/x). This is a useful identity to remember and can be easily visualized by drawing a right triangle and labeling the angles as arctan(x/y) and arctan(y/x) - 90 degrees. It is also important to note that this identity only holds true when both x and y are positive.