I am an engineering student and a test tomorrow requires us to take it without calculator. I actually know how to find arctan without a calculator: I draw a right triangle on the xy plane and see where the angle is located by referring to a trig unit circle.

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However, how do I find the arctan of a number that does not give a "clean" and rational result?

For example, how do I find the arctan of (4/5) without using a calculator? By using a calculator, the arctan of (4/5) is 38.66 ° or 0.674 in radians. How can I find this value without a calculator? I look at a lot of online resources but could not find help on this. Any help is appreciated! In practice, the answer is "you don't". If your exam forbids calculators, you will not need to calculate such values. In fact, many math professors will take points off if you give an approximate answer like 0.674 rather than writing arctan(4/5).

In principle, you can just do what the calculator does: use a Taylor series or an algorithm like CORDIC to approximate the value to the desired level of precision. However, this requires far too much computation to do by hand in a reasonable amount of time.

so should I just stick to drawing a right triangle on the x-y plane and refer its angle to a unit circle? I guess he will give us "perfect" angles.

Later in your engineering education, you will be leaving entire integrals in your final answer. They're getting you used to final answers that aren't just numbers.

If you know how to find the arctan of a "perfect" number, good. Be able to recognize when you have a "perfect" number and when you don't. If you do, give the exact solution. If you don't, leave your answer as arctan.

It's like when you're using logarithms and have, say, common log of 1000. You know this is 3 (since 10 cubed is 1000). But what's common log of 11? Who knows. Leave it as log of 11 and move along.

I think what’s important is knowing

tan(theta) is the slope of the terminal ray of theta, when the initial ray is at the 3 oclock position and the terminal ray is rotated counter clockwise from the initial ray.

With this idea your teacher can contruct a lot of problems.

Then for getting arctan, if you know the slope. You’ll need to know some special triangles, and remember that the range of arctan is (-pi/2, pi/2), and the period of tangent is pi, so that

Arctan(tan(x))= arctan(tan(x+ Npi)

If you can provide some more specific examples that your teacher gives I could tell you the trick they may want you to use.

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The class I am taking is called "Signals and Systems". For some problems, I have to calculate output of the signal when given an input signal. I do this by using the frequency response function given by the system.

In physics, the output signal is in the format y(t) = |A|*H(w) Cos(wt + theta) . I have to calculate theta from a complex fraction. Here is a sample problem that we have to know how to work:

https://imgur.com/a/VVyt9

The frequency response function is given, it is (1)/(3+jw). To calculate theta in the problem I linked, I have to do arctan(4/3). If I had a calculator, I would know it is 53 degrees. Only problem is, my professor is an old school, hardcore, "pulling weeds out", engineering professor. No calculators allowed. Each problem correct is worth 10 points. Each problem that is not correct, attempted, and showed reasonable logic is worth 5 points. So you could see why I would want to be able to calculate arctan(4/3) or any arctan value without a calculator :(