A lot of students are having trouble figuring out/remembering radians on the unit circle. Describe how you think about/how you figured out/ how you remember the radian measurements on the unit circle.
BE VERY SPECIFIC! Pretend you are teaching someone who has never heard of radians. Remember to be additive not repetitive.
To begin, radians are measured in terms of Pi. For example, one FULL circle (360 degrees) is equivalent to 2pi radians. Logically, this means that half of a circle (180) is 1pi radian.
ReplyDeleteThose are the basics, but it gets more complicated. For example, a question might ask you to find the angle measure of pi/4. So, take Pi (180) and divide it up into 4 equal pieces. Each angle should measure 45 degrees. The question can also be seen as 1Pi/4, meaning that it only wants you to take 1 chunk of those 4 (45 degrees). So your answer is 45 degrees. Also, a question might ask for 5Pi/3. As soon as you see the number in the numerator is larger than that of the denominator, the angle will be past 180 degrees, or Pi. 5Pi/3 is asking you for 5 chunks of 180 broken up into 3 pieces (60 degrees). 60+60+60+60+60= 300 degrees. Those are the basics.
ok so the first 4 are simple
ReplyDeletePi/6 Pi/4 Pi/3 Pi/2
those are the bases
now think of the circle as symmetrical.
since pi/3 was 60 then the first one in quad 2 is going to have a denominator of 3.
the 2nd quad is always going to be 1-the denominator. so the first one in quad 2 strating from the top would be 2Pi/3.
then it would be 4-1=3 so next would be 3Pi/4
then 6-1=5 so 5Pi/6.
then at 180 it will just be Pi
starting from the 3rd quad would be the denominator of 6. because the last one in quad 2 was 6. soooo the rule for the 3rd quad is
denominator+1. so the first one would be 6+1=7. so 7Pi/6. then 4+1=5 so 5Pi/4 then 3+1=4 so 4Pi/3.
ok the rule for the 4th quad is
2times the denominator-1.
so since you ended with the 3 in quad 3 you will start with 3 in quad 4.
so 3x2=6 6-1=5 so 5Pi/3
then 4x2=8 8-1=7 so Pi7/4
and last 6x2=12 12-1=11 so 11Pi/6
i hope this helps. and if you look at the unit circle while reading this it will make more sense and you will see the pattern better.
I been working on this too and i remember the first radians easily but the other ones i have trouble with. but i do remember that when the degrees end with 5 such as 45 135 then it has something to do with pi/4 then i see how many pi/4 i need to get to get to that number also a trick is to look at the graph circle if you look at it on the circle if it's pi/6 on the first Q then in the second then are across from each other and like misbah said you subtract what ever pi is but in the Q those you have to memorize. that's how i remember the radians
ReplyDeleteThe first things to remember are that Pi= 180 degrees of a circle and that 2pi= 360 degrees or a full circle. You can think of radians like pieces of 1 pi (remember 1 pi equals 1/2 of a circle or 180 degrees).
ReplyDeleteSo, when someone asks you about pi/2,pi/3,pi/4,and pi/6, simply divide the half of the circle into the denominator's given number (in equal pieces). Or think of it like 180/x. An easy way to practice this is to make the unit cirlce with just radians if your primary goal is to practice radians and to write the angle that corresponds with each radian. Key things to remember are:
pi/3= 60 degrees
pi/4= 45 degrees
pi/2= 90 degrees
pi/6= 30 degrees
Knowing the basic rules for the first quadrant are what help you to know the rest of the unit circle.
Okay so what you have to know is the 1st quad, if you know that then it would be easy for you to figure out the rest. So picture the 1st quad, and in it are 3 lines [well technically 5, if you include the x and y axis]The x axis would be 360 deg or 0 deg, which is 2pi. OH cuuhhrap! i forgot the most important thing to know!
ReplyDelete**PI=180 DEGREES** [very imp to know!] Anyways, so the x axis, or 0/360 degrees would be 2pi because 180 x 2= 360!
So forget that for now...lets focus on the three lines. so going upwards, after the y axis, that first line angle would be 30degrees, the next angle: 45 degrees, and the next 60 degrees, and the last, [y axis] is of course....90degrees! So we know that ...what equals 180 degrees??
answer: PI!!
So, 30 degrees would be pi/6 because 180 divided by 2 is 30! and so on...
pi/4= 180/4= 45.
pi/3= 180/3= 60.
pi/2= 180/2= 90.
This is a must -have-to-know-kinda-thing! just memorize it!
An easier way to remember this is...start out with the y axis[90 degrees] and go down... 2, 3,4, and 6. [just remember to add pi ..hehe]
so if you the first quad then it'll be no trouble at all to find out the rest!
Honestly, I just did what you told us to do. I memorized the first quadrant, and then I applied it to the other 3 quadrants and paid attention to if the x's or the y's or both were positive or negative. For the coordinates of the first quadrant, I just did 1,2,3 from up to down, then 3,2,1 going up next to them. I put them all over 2, then put a square root on every numerator except 1. So the order for the top is 1-3,2-2,3-1 without the denominator or the square root.
ReplyDeleteFor radians, I just knew that 180 divided by 6 is 30, 180 divided by 4 is 45, and 180 divided by 3 is 60 so thats how I remembered the degrees, then the small numbers are next to pi. I hope that makes sense, I tried to condense it my best! If you just remember the first quadrant, then you're good to go.
pi/3= 60 degrees
ReplyDeletepi/4= 45 degrees
pi/2= 90 degrees
pi/6= 30 degrees
If you know this,everything else comes easily. When dealing with different quadrants, just make sure you kno which values are positive or negative. Remember, "All Students Take Calculus".
Well, the way I learned to memorize and understand radians is that first of all a radian is the standard unit of angular measure and it is used in many areas of mathematics. 180 degrees is 1PI, so that is half of 360(a full circle) which means 360 would be 2PI. Now it would be putting pieces together, since you know Pi is 180, you can figure out pi/6 pi,4 pi/3=30dg,45dg,60dg. That is because it would be simple division dividing 180/6 and so on, and after that you would just remember that. When referring to graphs, there is a method "All Students Take Calculus" to memorize which quadrant sin,cosine, and tan are. Quad1=ALL, Quad2=Sin, Quad 3=Tan, Quad4=Cosine. Now when you want to know all the radians around the unit circle, it is basically memorizing. Other important points are, in the 1st quadrant all points are positive, in the second quadrant the x values are negative and the y's are positive. In the third quadrant both X and Y are negative, and in the 4th quadrant the y values are negative and X are positive. Hope this helps.
ReplyDeletepersonally i remember the radians as fractions of the half circle. the bottom number is how many ways you divide the semi-circle and the top number is the number of sections that you take. so 3π/4 is dividing your semi-circle into 4 sections then taking 3 of them.
ReplyDeleteWell to me all everybody needs to remember is that pi is a semi circle pretty much. And a semi circle is equal to 180 degrees. A whole circle is equal to 2pi, or 360 degrees. So yeah what you need to do is figure out how much percent the degree is out of 180 or 360. And convert it to a fraction. That's what i be doing anyways.
ReplyDelete