Hey everyone, I've got a question about finding the basic sines/cosines of the unit circle, without relying solely on pure memorization. I've got them all memorized personally, so it's not really an issue for myself, but I'm currently tutoring someone in trig, and they're having a lot of trouble memorizing the first quadrant of the unit circle. I vaguely know about this method, where if we write the basic angles from the first quadrant in this form, matched up respectively with the corresponding fraction in the series below the given angle-
[tex]0 \ \ 30 \ \ 45 \ \ 60 \ \ 90[/tex]
[tex]\frac{0}{4} \ \frac{1}{4} \ \frac{2}{4} \ \frac{3}{4} \ \frac{4}{4}[/tex]
Then we can take the corresponding values, and find that sine/cosine.
For example, for [itex]\sin45[/itex], we'll use [itex]\frac{2}{4}[/itex], which then simplifies to [itex]\frac{1}{2}[/itex], then we'll simply take the square root of that value, and we have[itex]\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}[/itex], which is the sine of 45 degrees. The same process will work for any of the angles. To find [itex]\sin60[/itex], we'll take [itex]\frac{3}{4}[/itex], then take the square root to get [itex]\frac{\sqrt{3}}{2}[/itex] which is the sine of 60 degrees.
To find the cosines of the respective angles, you simply reverse the order of the fractions into descending order, and use the same process.
Are there any problems with this method? Obviously it is only going to work for the base values from the unit circle, but for those values it does seem to be quite effective. I've searched, and haven't managed to find anything in the way of reading about it. I'd like to try explaining this method to the person that I'm tutoring, in hopes that this will help them find these values, in lieu of having a unit circle. I want to make sure I fully understand it before explaining it to them though. It's really pretty intuitively simple, so I'm hoping it will be easier for them than simply trying to memorize it.
Is there a way to apply a similar method to the values of the other quadrants, or does that still rely on being able to reflect the angles into the different quadrants? I can see it being possible by selectively using negative signs, but that seems like it would be harder to memorize than memorizing how to reflect the angles into different quadrants.
[tex]0 \ \ 30 \ \ 45 \ \ 60 \ \ 90[/tex]
[tex]\frac{0}{4} \ \frac{1}{4} \ \frac{2}{4} \ \frac{3}{4} \ \frac{4}{4}[/tex]
Then we can take the corresponding values, and find that sine/cosine.
For example, for [itex]\sin45[/itex], we'll use [itex]\frac{2}{4}[/itex], which then simplifies to [itex]\frac{1}{2}[/itex], then we'll simply take the square root of that value, and we have[itex]\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}[/itex], which is the sine of 45 degrees. The same process will work for any of the angles. To find [itex]\sin60[/itex], we'll take [itex]\frac{3}{4}[/itex], then take the square root to get [itex]\frac{\sqrt{3}}{2}[/itex] which is the sine of 60 degrees.
To find the cosines of the respective angles, you simply reverse the order of the fractions into descending order, and use the same process.
Are there any problems with this method? Obviously it is only going to work for the base values from the unit circle, but for those values it does seem to be quite effective. I've searched, and haven't managed to find anything in the way of reading about it. I'd like to try explaining this method to the person that I'm tutoring, in hopes that this will help them find these values, in lieu of having a unit circle. I want to make sure I fully understand it before explaining it to them though. It's really pretty intuitively simple, so I'm hoping it will be easier for them than simply trying to memorize it.
Is there a way to apply a similar method to the values of the other quadrants, or does that still rely on being able to reflect the angles into the different quadrants? I can see it being possible by selectively using negative signs, but that seems like it would be harder to memorize than memorizing how to reflect the angles into different quadrants.
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