Consider our table of values that we have started to generate for the sines and cosines of certain angles. So far we have 0, 30, 36, 45, 60, 90 and we can surely find more based on these. However, we then have all the other angles (infinitely many to be precise) for which we do not know sines and cosines and are concerned about how we could generate these values using geometric approaches.
You have been given a formula sheet that consists of relationships that the Muslim astronomers and their Indian and Greek predecessors were well aware of. Although we may not use these without the necessary derivations that we will undertake soon, we can get an idea of the possibilities of angles for which sines and cosines could be determined based on these formulas.
So, list as many angles between 0 and 360 degrees that we can find sines and cosines for based on the formulas you have been given. Also, ponder the first two formulas and determine whether they are just stating what you had already observed before or are they completely new formulas for you. We will discuss this further in class.
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12 comments:
This may be a trivial question, but does that just imply plugging in a random sine to find its cosine? Thanks
i have the same question as Vranian. am i missing something deeper here?
Okay, so you need to be more specific with your question. Which formula are you referring to? Plugging in random sines won't help if you don't have the lengths of the sines you are plugging in. You can only plug in sines and cosines that you know the lengths of based on the work we have conducted so far. That would streamline it to about 6 angles that we have computed sines and cosines for. If you had the formulas I have given you, which other could you find?
What, are we supposed to also find more sines off of the sines we found in the formulas? Such as, I got 150 as another sine we could find for one of the things. Would I have to plug in sin(150) and find more sines?
Well, remember that I am not asking you to actually FIND them. But if you did plug in 150 degrees into one of the formulas, say the sine subtraction formula, then you could use 150 and, say, 36 degrees and get the sine of 114 degrees. Aha! so another angle for which it is possible to compute a sine length and, hence, a cosine length. Like this keep going and keep listing angles that you can find sines and cosines for (without actually finding them).
That's what I had meant. I was just checking... this might take longer than originally planned...
Well, not really! You'll find that there is a limitations on the angles you'll be able to find. Also, you can keep it to whole numbered angles between 0 and 360 for now. You can pretty much sit with a calculator and start jotting down angles.
Steven and Chap, let me know if you're clear on what's being asked.
Julia said...
So, can't you basically find any angle between 0 and 360? What would a limitation be?
haha oops. I did that post wrong.
So, you mean you found some limitations?
Well, here is a question I'll pose to you then:
Can you find the sine of 20 degrees?
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