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Shalashaska_123
United States142 Posts
I've written a solution to your problem. Unfortunately, I can't post the .pdf file here, so I had to export it to two lesser quality image files. Here they are. ![[image loading]](http://i.imgur.com/lWIDU4C.png) ![[image loading]](http://i.imgur.com/iPuVPN0.png) In addition, here are some other trigonometric identities I think you'll find useful in the future. ![[image loading]](http://i.imgur.com/UhKojck.png) Good luck on your final. Sincerely, Shalashaska_123 EDIT: Fixed some mistakes.  | ||
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Maenander
Germany4919 Posts
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coverpunch
United States2093 Posts
In US colleges, stuff like this also tends to be a way of weeding out untalented or unmotivated students. If you can't derive this using the other identities and/or you don't want to try, then you don't have the ability or work ethic to succeed in a mathematical career. Both because the material gets much harder when you hit analysis, and it's masochistic to try if you don't know how these fundamental things are related to each other. I am curious about something. American math books tend to have explanations with "proof is left as an exercise to the reader". I'll assume the big names for standard textbooks are the same in all English speaking countries. Do they do this with math books in other languages? I would tentatively guess yes, having studied books translated into English by German or Russian authors and they do it too, but I'd just like to know from a person who actually uses books in other languages. | ||
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Ghostcom
Denmark4781 Posts
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munch
Mute City2363 Posts
On May 18 2015 20:06 Maenander wrote: Nice write-up, but sin(x) = cos(pi/2-x) is pretty much evident from the definitions of sinus and cosinus, so I don't know what's to discuss. You can say that for everything in Maths. Everything is trivial to someone who understands it thoroughly. The important part here is teaching bulletproof logic. | ||
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Cambium
United States16368 Posts
sin x = b /c cos ( .5 pi - x ) = b /c :.sin x = cos ( .5 pi - x ) | ||
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Grovbolle
Denmark3804 Posts
On May 18 2015 20:06 Maenander wrote: Nice write-up, but sin(x) = cos(pi/2-x) is pretty much evident from the definitions of sinus and cosinus, so I don't know what's to discuss. Exactly | ||
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Shalashaska_123
United States142 Posts
EDIT: Also, if anyone would like to have more practice with these kinds of problems, try proving the identities in the right column of the handout I posted with the identities in the left column. On May 18 2015 21:38 thecrazymunchkin wrote: You can say that for everything in Maths. Everything is trivial to someone who understands it thoroughly. The important part here is teaching bulletproof logic. No, Maenander is right. As Cambium eloquently wrote, On May 18 2015 22:24 Cambium wrote: sin x = b /c cos ( .5 pi - x ) = b /c :.sin x = cos ( .5 pi - x ) it is quite simple to show that sin x = cos ( .5 pi - x ) from the definitions of the functions. I wouldn't go so far as to say there's nothing to discuss, though. Clearly there are some people, such as travis, that are confused and need some explanation. That's where I think you are right. On May 18 2015 22:24 Cambium wrote: stealing this image: Oh, are you in the same class as travis? That's nice I was able to kill two birds with one stone. Show my second page some love, too!On May 18 2015 20:40 coverpunch wrote: It sticks better and represents a grasp of the intuition if you're able to derive it algebraically. Yes, I agree. On exams especially, where you need to demonstrate your knowledge to the professor, the geometric reasoning isn't nearly as impressive. One should ideally be able to think about these problems both algebraically and geometrically. On May 18 2015 20:40 coverpunch wrote: In US colleges, stuff like this also tends to be a way of weeding out untalented or unmotivated students. If you can't derive this using the other identities and/or you don't want to try, then you don't have the ability or work ethic to succeed in a mathematical career. Both because the material gets much harder when you hit analysis, and it's masochistic to try if you don't know how these fundamental things are related to each other. Analysis is definitely the class that distinguishes the thinkers from the calculators in the mathematics curriculum. Unfortunately, what one is taught in grade school in the U.S. doesn't prepare students for this class--the emphasis is on computing. For example, you're given the Pythagorean Theorem or Law of Cosines and asked to compute sides of triangles. You're told the results of theorems and asked to compute a derivative or solve differential equations with integration or series solutions without any questions about the theorems themselves. Most if not all of the questions in grade school involve lots of tedious calculations with little to no creative thinking on the part of the student. Mathematics is really concerned with "why" as opposed to "how" things work, and it's a shame that students (at least mathematics students in the U.S.) don't get a good idea of this early on. Anyway, pardon my little off-topic rant. I just wanted to say what was on my mind. | ||
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Deleted User 3420
24492 Posts
On May 19 2015 03:31 Grovbolle wrote: May I ask how old you are and the level at which this is taught? Exactly I am 30 years old, I am being taught this at community college. I had to start with "intermediate algebra", which was a pre-req class. Then a pre-calculus class that was basically algebra 2, then this pre-calculus class which is basically trig. This class was an online class, which is really just code for self-taught, so my understanding is a lot weaker than the other 2 classes... because I did a pretty unmotivated job of teaching myself. | ||
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munch
Mute City2363 Posts
On May 19 2015 06:30 Shalashaska_123 wrote: No, Maenander is right. As Cambium eloquently wrote, it is quite simple to show that sin x = cos ( .5 pi - x ) from the definitions of the functions. I wouldn't go so far as to say there's nothing to discuss, though. Clearly there are some people, such as travis, that are confused and need some explanation. That's where I think you are right. You're right, I didn't really express myself correctly. It would be more accurate to say that nothing is trivial to someone who doesn't understand it, which is why I think that dismissing it as 'evident' in this case was a little counterproductive. In any case, hopefully we've got there in the end. Do you teach in the States then? I feel we agree on quite a lot of the flaws of the maths teaching system; nice (or not nice, really) to know that it's not just a British problem! On May 19 2015 10:27 travis wrote: I am 30 years old, I am being taught this at community college. I had to start with "intermediate algebra", which was a pre-req class. Then a pre-calculus class that was basically algebra 2, then this pre-calculus class which is basically trig. This class was an online class, which is really just code for self-taught, so my understanding is a lot weaker than the other 2 classes... because I did a pretty unmotivated job of teaching myself. If you need some stuff explaining feel free to give me a shout anytime | ||
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doubleupgradeobbies!
Australia1187 Posts
2*cos(x)cos(pi/2-x) = 2*((e^ix +e^-ix)/2)) * ((e^(pi*i/2 - ix) + e^-(pi*i/2-ix)/2)) = (e^ix +e^-ix)*((ie^-ix - ie^ix)/2) = (e^i2x - e^-i2x)/2i = sin(2x) I know alot of professors don't like to accept answers outside the scope of the course, but seriously, how easy is that? edit: oops was originally proving the wrong thing | ||
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Geiko
France1933 Posts
On May 18 2015 09:01 travis wrote: I am reviewing for my trig final (THIS ISNT AN ASSIGNMENT, IT IS REVIEW). Two of the questions in the review aren't really something that was directly covered in the class, and I am having trouble understanding it and also trouble researching it myself. The problem says: I don't really understand what my professor is asking me to do. I mean, I know sin(2x) = 2 cos(x)(sin x) so then i guess sin(x) = cos(pi/2-x) ? But I mean I don't really know how to prove that, or what my professor wants me to *show* on the right triangle. I could show any given 2 angles that add to 90 on the right triangle and label them x and (pi/2-x) but wtf would the point of that be? P.S. I'll delete this blog after some time or after I get help because I know how strict TL can be about shitty blogs This is how I understood the problem. You have to show geometrically, using right triangles that the above formula is correct. ![[image loading]](http://i.imgur.com/3AelcUc.jpg) This geometrical "proof" (can't prove anything with drawings) uses: Center angle is twice as big as angle from a point on the circle Triangles with points on a circle and one side as a diameter are right triangles In all triangles, sin(Â)/BC=sin(B^)/AC=sin(C^)/AB Sum of angles in a triangle is pi | ||
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radscorpion9
Canada2252 Posts
For people using Euler's formula that might need to be proven using a power series as well, otherwise there's no good reason to accept the formula (it is the foundation for one of the most famous equalities in math after all, it probably deserves an explanation). Given that they probably are far far away from power series I don't think its reasonable to use such proofs at that level. | ||
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corumjhaelen
France6884 Posts
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Bannt
United States73 Posts
For people using Euler's formula that might need to be proven using a power series as well, otherwise there's no good reason to accept the formula (it is the foundation for one of the most famous equalities in math after all, it probably deserves an explanation). Given that they probably are far far away from power series I don't think its reasonable to use such proofs at that level. Yes, it's a good assumption that you should never use something outside of the requirements to take the course and the course itself in math. The instructor wouldn't know whether you have covered the material or you found a formula and blindly applied it. And using things that you don't understand kind of defeats the point of the courses, and in a way, math itself. | ||
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coverpunch
United States2093 Posts
On May 20 2015 05:34 radscorpion9 wrote: Another question is whether you need to prove the double-angle formula, or any other sum-difference formulas that many people used in their proofs - or are they expected that you memorize and use them? In which case a proof like what Geiko did is necessary, although he did use the law of sines. Also he didn't completely finish at the end; just convert sin(pi/2-a) = cos(pi/2 - pi/2 +a) = cos(a) and sin(a) = cos(pi/2 - a) completing the proof. For people using Euler's formula that might need to be proven using a power series as well, otherwise there's no good reason to accept the formula (it is the foundation for one of the most famous equalities in math after all, it probably deserves an explanation). Given that they probably are far far away from power series I don't think its reasonable to use such proofs at that level. From my experience in American math classes, at the early levels they'll give you a sheet of identities like the one Shalashaska provided and you're expected to "plug and chug" as they say. From the OP's subsequent posts, it appears cos(pi/2-x) = sin(x) was not one of the identities he was given so he was expected to prove it algebraically and show it geometrically. It's only after you demonstrate a firm grasp of the basics and move on to basic analysis courses or other courses developing proofs that you'll go back and do proofs for the identities. I did my Master's in mathematical statistics and one memorable problem was going back and doing a proof for Hardy-Weinberg equilibrium (that in a population with two alleles and random mating, every generation will have genotypes in ratio AA:2Aa:aa). In high school biology, you learn to use Punnett squares with Mendel's laws and see the result with counting. It's much trickier to derive the result algebraically and generalize it to any number of subsequent generations. But at that level, it was a useful exercise to teach statisticians how to approach problems in other subjects from a mathematical and theoretical perspective. | ||
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Shalashaska_123
United States142 Posts
On May 20 2015 05:34 radscorpion9 wrote: Another question is whether you need to prove the double-angle formula, or any other sum-difference formulas that many people used in their proofs - or are they expected that you memorize and use them? In which case a proof like what Geiko did is necessary, although he did use the law of sines. Also he didn't completely finish at the end; just convert sin(pi/2-a) = cos(pi/2 - pi/2 +a) = cos(a) and sin(a) = cos(pi/2 - a) completing the proof. For people using Euler's formula that might need to be proven using a power series as well, otherwise there's no good reason to accept the formula (it is the foundation for one of the most famous equalities in math after all, it probably deserves an explanation). Given that they probably are far far away from power series I don't think its reasonable to use such proofs at that level. radscorpion9, you don't have to re-invent the wheel to write an acceptable proof. The sine double angle formula and the cosine difference formula that I used are presented as theorems and proven in any precalculus text. These are elementary, and you don't have to prove them every time you use them. There are better things to do with your time; in addition, it takes the focus away from the problem at hand. If you happen to forget them and don't have a reference like the handout i provided, then yes, you'll have to re-derive them. The same goes for the Law of Sines. I'll briefly go over the proofs in case you are curious.. Sine Double Angle Formula from Sine Sum Formula sin 2x = sin (x + x) = sin x cos x + cos x sin x = 2sin x cos x Sine Sum formula from Cosine Difference Formula sin (a + b) = cos ( pi/2 - a - b ) = cos( pi/2 - a )cos( b ) + sin( pi/2 - a )sin(b) = sin a cos b + cos a sin b Outline of Cosine Difference Formula Proof Consider two points P_1 and P_2 on a unit circle that have angles a and b, respectively, with respect to the x-axis. Rotate the circle clockwise by an angle b so that the points that were at P_1 and P_2 are now at the new points P_3 (with angle a - b with respect to x-axis) and P_4 (with angle 0 with respect to x-axis), respectively. The distance between P_1 and P_2 is equal to that between P_3 and P_4. Simplify this equation with tricks from algebra, and the formula, cos (a - b) = cos a cos b + sin a sin b, pops out. Your second paragraph is nonsense. Of course there are good reasons to accept Euler's formula. If you're faced with the task of simplifying a nasty trigonometric expression that even that handout can't help you with, it's often far more convenient to convert everything to exponentials because of their handsome algebraic properties. Once you're done working with exponentials, switch back to trigonometric functions as doubleupgradeobbies! did in his or her proof. On May 19 2015 15:49 doubleupgradeobbies! wrote: 2*cos(x)cos(pi/2-x) = 2*((e^ix +e^-ix)/2)) * ((e^(pi*i/2 - ix) + e^-(pi*i/2-ix)/2)) = (e^ix +e^-ix)*((ie^-ix - ie^ix)/2) = (e^i2x - e^-i2x)/2i = sin(2x) As I said in my previous post, the education system emphasizes computation rather than proofs. Students don't learn mathematical induction and prove De Moivre's formula before they start using it to find complex roots. Similarly, students don't prove Euler's formula with Taylor series from calculus before they start using it to manipulate trigonometric expressions. | ||
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munch
Mute City2363 Posts
On May 19 2015 10:27 travis wrote: I am 30 years old, I am being taught this at community college. I had to start with "intermediate algebra", which was a pre-req class. Then a pre-calculus class that was basically algebra 2, then this pre-calculus class which is basically trig. I highly doubt that he's got to complex numbers yet. There's no point in not keeping it simple for now | ||
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Shalashaska_123
United States142 Posts
On May 20 2015 16:04 thecrazymunchkin wrote: To quote the OP: I highly doubt that he's got to complex numbers yet. There's no point in not keeping it simple for now Complex numbers are covered in Algebra 2 here in the U.S., but you're right. It's always best to keep it as simple as possible. I was just mentioning the exponential approach above in case a much harder problem than the one dealt with here was encountered. With that being said, On May 20 2015 06:53 corumjhaelen wrote: sin x = cos ( .5 pi - x ) from the definitions of the functions being obvious clearly depends of the definitions. Have fun doing that from the power series. Hard or easy rarely is clear cut. use the SOH-CAH-TOA definitions. Why would you make life difficult for yourself using series representations of the functions? Taylor series shouldn't even come to mind when dealing with such a simple problem in trigonometry. They wouldn't even help a trigonometry student either. | ||
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