The pre-quiz here really gives it away: you need to be able to recognize any radian measure as a multiple of π/6 or π/4, and any degree measure as a multiple of 30° or 45°… and yes, be able to convert from one to the other quickly and accurately.

Let’s be clear that neither degree measure nor radian measure is a great system for measuring circles. In my version of a perfect world, every angle would just be a fraction of a circle. You’d ask me: what’s a right angle? And I’d say, that’s a quarter circle, of course. 45 degrees? I’d call that an eighth of a circle. 60 degrees? A sixth of a circle. And so on.

But that’s not our world at present. Instead, for the purposes of this test, you need to know degree measure and radian measure.

So, first things first: I’m going to assume that you’re comfortable with degree measure, at least for the multiples of 30° or 45°, and at least a little bit comfortable with radian measure.

Next, let’s cut straight to the good news: all you really to know is that there are 360 degrees in a circle, and there are 2π radians in a circle. That’s because just like every angle is a fraction of a whole circle, each angle is also the same fraction of 360 degrees or 2π radians, your choice.

Examples:

• You say 60 degrees, I see a sixth of a whole circle, so I know it’s in the first quadrant near the y-axis. If I need to convert to radians I say, okay, it’s still ⅙ of a circle, so that means ⅙ of ...2π radians. That’s π/3 radians.
• You say 45 degrees, I see an eighth of a circle, so I know it’s in the first quadrant halfway between the two axes. If I need to convert to radians, I say, ⅛ of a circle is ⅛ of 2π radians. That’s ...π/4 radians.

Whether you use my system or not, you do need to be able to identify and see all angles that are multiples of π/6 radians or π/4 radians comfortably, fluently, automatically. So for example, if I say picture an angle that’s 5π/6 radians, I need you to know that it’s in the second quadrant, 30 degrees above the negative x-axis. And I need the image of that angle to appear in your head nearly as quickly and easily as if I had said picture a 90-degree angle (which by the way, this isn’t) or a 150-degree angle (which, you need to know, it is).

Of course, flashcards are likely to be very helpful here.

Now, worst case, nothing in this section works for you, and you’re saying to yourself, I have to do this with a calculator.

That’s not my top choice for you, but if that’s what you have to do, then fine. In that case, though, just practice enough that you can do the conversions in both directions quickly, accurately, and effortlessly.

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