This assignment is to be done on seperate sheets of paper as it will be collected on Monday.
1. Using only the formulas we’ve derived so far and the computations of sines of angles we have done, determine the value of sin(3°). (Use your algebraic skills at first and then use your calculators to get a numerical answer.)
2. Read the article on Jamshed al-Kashi.
3. Go back to where the formula involving sin(3 theta) occurs in the article and then attempt problem 4 of this assignment.
4. Using the sine and cosine addition and subtraction identities, prove the formula .
5. Use fixed-point iteration, as outlined in the article, to get a value for sin(1°) to about 6 decimal places of accuracy.
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This is in response to Victoria's question. For now I shall use x instead of theta since i don't have that symbol available to me when commenting.
For number 3 on the assignment, my first hint is to get an expression for sin(2x+x) using the addition formula. Within this formula you can replace one of the terms using the double angle formula and another using the cosine addition law for cos(x+x).
Some algebraic manipulation will be required and make sure not to skip steps. Lastly, make use of the pythagorean relationship between sine and cosine to replace any cosine terms once you get an expression entirely in terms of sine and cosine of x (without any 2x showing up in there).
In number 5, are we using a graph to find sin(1), or are we using a different method? I understand what the article was saying about the graph, but I am not sure how exactly to go about doing this. Any suggestions?
I've gotten an expression for (2x+x) and replaced the terms using the double angle formula and the cosine addition formula.
However, I'm having trouble with using the pythagorean relationship between sine and cosine to replace the cosine terms in my expression.
Scratch my last post!
The reason why it wasn't working was because of a silly mathematical error earlier on in my work.
Just as a side note... It helps to take number 3 slowly so you don't mess up the algebra.
Now to provide some helpful tips for number 5.
Notice that the article solves the equation you derived in number 4 for sin(1*). The next strategy that the article suggests is to use a variable, say x, for the sin(1*) and re-write the equation in terms of x.
Then, it should also be easy to see that solving for the X the way the article suggests is solving for sin(1*), except that there still a (sin(1*))^3 on the right side of the equation. That X^3 is our guessing place and the X we've solved for will be our next guess.
more in next comment...
Start with a number that you think is close to the value of sin(1*). The article suggests that you start with (sin(3))/3 and that is a pretty good suggestion. So you plug in the (sin(3))/3 into the right side of the equation, for X, where the 4X^3 occurs. Then find a value for the right side of the equation and this becomes a value for X. This also is the value for your next guess. So plug in what you found back into the right side of the equation and then you'll get another value for X on the left side.
Repeat this process and you will see that each new value will start to come closer and closer to the value of sin(1*).
Ohh... I see now. Thank you.
Do we use our calculators for number 5?
i'm still confused on #1. i think to find the sine of 3 you can use the sine subtraction of 36-30 then use the sine half angle formula to find sine of 3 but i don't think i am doing that right. it may be because i didn't simplify the sine of 36 right but i don't know how to do that either
Well, then you need to go back and fix the sin(36*) calculation. You can check whether you got the right expression for it by checking the final value with the calculator's sin(36) [make sure you are in degree mode though].
You are correct about your procedure for finding the sin(3) though.
the answer to Emily's last question is YES
I can't get number 1 either. My calculation for sin(36) is wrong and I can't seem to get the right one.
I have (this is a a square root sign): (√1-((1+√5)/4)^2) Sorry..I didn't know how to put it down..
For number 3, I'm trying to use the formulas that were recomended, but am having trouble first grasping the concept of the formula we're trying to prove. Can somebody give me some bearings possibly?
I have the same problem...
Scratch my post...
One last thing, for #5... Do we just keep plugging in the values for x seven times to get the accuracy we need? Is that what's the problem is asking?
I am confused on 3. what do you mean by an equation for sin(2x+x)? im not sure how to find that using the addition formula.
I was just finishing up number five and after the first two times I've plugged in a new value for x, I keep getting the same ending value as the last... Is there anything I'm missing?
Ah, and another thing I just realized; my value of (sin(3))/3 was closer to the sin(1) than the value of X for when I had plugged in (sin(3))/3...
For number three I ended up with a negative expression for sin(alpha).... Maybe I messed up when substituting but can anyone elaborate on how I might fix it?
for number 5, is there only a need for two "tries," or do I not fully understand?
For #5 are we simply supposed to use the equation in the article and our calculator to get the value of sin(1)??
victoria, it only took me two tries as well so i'm guessing we did the same thing. but i'm not sure if that is all that was needed
I only did it twice also
If two tries get you the 6 decimal places of accuracy then fine. If not, then do another try. Remember that your calculators can only give you about 9 decimal places of accuracy. So, now you can see why Al-Kashi's achievement for that time period and to do it without a calculator was such a feat.
Moreover, if you kept doing more tries, you'd get an accuracy that is even better than you calculator. But for that you would have to use a computer software that can generate a greater accuracy than your calculators.
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