Commonly Encountered Angles
Commonly Encountered Angles
This chart up here on the board is a list of commonly encountered angles. When you first glance at this list it may look a little bit overwhelming because there’s so many numbers and letters, so let me break it down for you. Up here at the top we have 2 rows talking about angle measures, so let’s go ahead and dive into that.
For 0 degrees that’s equal to 0 radians, 30 degrees is equal to Pi over 6 radians, 45 degrees, Pi over 4 radians; 60 degrees, Pi over 3 radians; and 90 degrees is equal to Pi over 2 radians. 0 degrees is easy to remember because it’s equal to 0 radians, and then you start out with 30 degrees, so the key here is remembering that 30 degrees is equal to Pi over 6.
From there, as you progress to the next percentages, notice that the denominator just decreases from there. We start out with Pi/6, then we go to 45, it’s Pi over 4, then 60 is Pi over 3, then 90 is Pi over 2. Now we do skip Pi over 5, but if you can kind of remember that Pi’s are always going to be the numerator and the only thing changing here is the denominator that should be helpful to you.
Now from here we have 3 columns, and in each column, we have a trigonometric function and then it’s reciprocal. Here we have sine and its reciprocal cosecant, then we have cosine and its reciprocal secant, and then we have a tangent and its reciprocal cotangent.
We have these trigonometric functions coupled with these different percentages and we’re getting what those are equal to. The reason I have these up here is these are the simple angles, these are ones that you’re going to come across a lot, because these numbers are all multiples of 15, a lot of them end in 0, (so you’re going to come across them a lot) and they’re fairly simple numbers.
If you look at this chart you can kind of cut it in half because you have these basic 3 trigonometric functions, (sine, cosine, and tangent) and if you go to where those stop, right before their reciprocals start, if you were to draw an imaginary line right down the middle of this chart, everything up here is what you need to be very familiar with.
You want to be familiar with all of this, but if you’re really going to study something really hard, if you’re really going to memorize something, it’s going to be this stuff up here because that’s what you’re going to come in contact with the most.
Your eventual goal is to hear, or just see, this right here without the answer and go, “Oh, sine 0 degrees that’s equal to 0, or cosine 45 degrees that’s the square of 2 over 2.” Now maybe your goal isn’t to be able to totally memorize it, but maybe just to be able to come up with it fairly quickly, because if you know the answers to these off the top of your head, that’s going to save you a lot of time when solving problems.
If you don’t have to plug these things into a calculator every time, because many problems you’re not going to really come across anything that’s not going to have these angles in it. It’s only in harder problems that you’re going to have harder angles.
I want to go ahead and go through some of these: sin 0 degrees we already said is equal to 0, then sin of 30 degrees is equal to 1/2, and sign of 45 square root of 2 over 2, sign of 60 the square root of 3 over 2, and the sign of 90 equals 1, so you may want to write these down as we go.
Since there’s so much information here I’m going to go ahead and not talk about the reciprocals. The information is up here, but like I said earlier the important stuff is on the top, so I’m going to focus on that. Now we transition to cosine: cosine 0 degrees is equal to 1, cosine of 30 degrees is equal to square root of 3 over 2, the cosine of 45 degrees is equal to the square root of 2 over 2, cosign of 60 degrees is equal to 1/2 or point 5, and the cosine of 90 is equal to 0.
Then we go to tangent of 0 degrees is equal to 0. Notice here at the top here we have 0 degrees all the way across, and the only one that has something (a value) other than 0 is cosine where it’s equal to 1. Then, tangent of 30 degrees is square root of 3 over 3, tangent of 45 is equal to 1.
The tangent of 60 is equal to square root of 3, and the tangent of 90 is undefined. Focus on memorizing, if anything, just the ones that are equal to 0, 1, or undefined. Like I said, remember all the 0 degrees. That sine and tangent 0 degrees are equal to 0, and the cosine of 0 degrees is equal to just 1.
The sine of 90 degrees is equal to 1, the cosine of 90 degrees is equal to 0, and then the tangent of 90 degrees is equal to undefined. Then we also have a 1 right here for the tangent of 45. If anything, just remember those right there. There are many different ways to remember these, there may be some tricks you can come up with.
Flashcards may also be a good way to try to remember all of these commonly encountered angles, but like I said, you want to be familiar with all of this information, but if anything, focus on the top half because those are the angles that you’re going to encounter the most.