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C2 - Logarithms

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Original post by Chittesh14
This example...


Ah right, the weird thing that happens with sine rule, I'll have a go now. Do you understand it?
Original post by RDKGames
Ah right, the weird thing that happens with sine rule, I'll have a go now. Do you understand it?


Idk lol. I got 1 answer right and the other I got wrong. The 2nd answer which I got wrong was in my working out - but it was for a different angle. Maybe I got confused, idk.


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Largest angles are 101 and 131. If you think about the sine rule, it only needs length and one angle in order to find the other. It does not specify any directions of lines hence why there are two different solutions geometrically as you can see. Also sin(A)=sin(180-A) so you can use that also in your sine rule with an unknown angle, and this would give you two different values for the angle. In the context of this question, it wants the largest of the angles, so you'd have to sketch two triangles and see which angle is the largest.


ImageUploadedByStudent Room1467137999.523532.jpg


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(edited 7 years ago)
Original post by RDKGames
Of course it had to be something as simple as that :/

tan(kπ20)=1+5−5+25[br]=1[br]=1+5+5+25[br]=1−5−5−25[br]=1−5+5−25 tan(\frac{k\pi}{20}) = 1+\sqrt5-\sqrt{5+2\sqrt5}[br]=1[br]=1+\sqrt5+\sqrt{5+2\sqrt5}[br]=1-\sqrt5-\sqrt{5-2\sqrt5}[br]=1-\sqrt5+\sqrt{5-2\sqrt5}

for k=1,5,9,13,17k=1, 5, 9, 13, 17 respectively.


I really like questions where you have to find exact values of trig functions. Something very satisfying about it.A while back now, I used similar methods to find the exact value of
Unparseable latex formula:

\sin \left (\frac{3\pi}{7} \right) \displaystyle = \sqrt{ \sqrt[3]{ \frac{7}{3456} \left (-1+3\sqrt{3} i \right ) } +\frac{7}{144} \sqrt[3]{ -\frac{864}{49} \left (1+3\sqrt{3} i} \bigr )+\frac{7}{12}}

.
Original post by Ano123
I really like questions where you have to find exact values of trig functions. Something very satisfying about it.A while back now, I used similar methods to find the exact value of
Unparseable latex formula:

\sin \left (\frac{3\pi}{7} \right) \displaystyle = \sqrt{ \sqrt[3]{ \frac{7}{3456} \left (-1+3\sqrt{3} i \right ) } +\frac{7}{144} \sqrt[3]{ -\frac{864}{49} \left (1+3\sqrt{3} i} \bigr )+\frac{7}{12}}

.


Some cheeky complex numbers in there, I'd like to see how they cancel out xD
Original post by RDKGames
Some cheeky complex numbers in there, I'd like to see how they cancel out xD


They don't.
Original post by Ano123
They don't.


How does that work? Wouldn't that mean sin(3pi/7) is a complex number itself?
Original post by RDKGames
How does that work? Wouldn't that mean sin(3pi/7) is a complex number itself?


It's weird, I can't really wrap my head around it. That's maths for you isn't it. Here's a page you may want to read if you're interested. Are you a further maths student - I assume so.
https://en.wikipedia.org/wiki/Casus_irreducibilis
Original post by Ano123
It's weird, I can't really wrap my head around it. That's maths for you isn't it. Here's a page you may want to read if you're interested. Are you a further maths student - I assume so.
https://en.wikipedia.org/wiki/Casus_irreducibilis


I can sort of understand why that happens, as they give the example with the cube root of 1; it has 2 complex conjugate roots, probably works similarly when it comes to the solution you've posted. The way they describe everything is quite beyond anything I've done previously so I can't fully wrap my head around it either but it makes sense for this to happen.

And yes, I'm a FM student.
Original post by RDKGames
Largest angles are 101 and 131. If you think about the sine rule, it only needs length and one angle in order to find the other. It does not specify any directions of lines hence why there are two different solutions geometrically as you can see. Also sin(A)=sin(180-A) so you can use that also in your sine rule with an unknown angle, and this would give you two different values for the angle. In the context of this question, it wants the largest of the angles, so you'd have to sketch two triangles and see which angle is the largest.


ImageUploadedByStudent Room1467137999.523532.jpg


Posted from TSR Mobile


Wow your method is short asf... And correct answers! I made a mistake - I thought it meant the two triangles that were formed when you split the large triangle. I got the answer right then but with some long asf working out lol.

Thanks :smile:
Posted from TSR Mobile
Original post by RDKGames
I can sort of understand why that happens, as they give the example with the cube root of 1; it has 2 complex conjugate roots, probably works similarly when it comes to the solution you've posted. The way they describe everything is quite beyond anything I've done previously so I can't fully wrap my head around it either but it makes sense for this to happen.

And yes, I'm a FM student.


There's a difference though with the cube roots of unity though, some of the cube roots of unity are complex, but with the exact value of sin⁡(3π/7) \sin (3\pi/7) it is purely a real number.
Original post by Ano123
There's a difference though with the cube roots of unity though, some of the cube roots of unity are complex, but with the exact value of sin⁡(3π/7) \sin (3\pi/7) it is purely a real number.


Then there must be a way they cancel out, perhaps just not possible to show by hand or something. One thing is for certain that we can agree on: that thing is NOT a complex number overall. Plus they are both complex conjugates of eachother so my money's on our inability to show them cancelling out.
(edited 7 years ago)
Original post by RDKGames
Then there must be a way they cancel out, perhaps just not possible to show by hand or something. One thing is for certain that we can agree on: that thing is NOT a complex number overall.


Yeah It is certainly not a complex number.
Original post by RDKGames
x


Part c please. Can you answer it and show me how you got to the answers thanks :smile:

Answer to the question: 36.8 degrees

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(edited 7 years ago)
image.jpeg113.0KB
image.jpeg134.6KB
Original post by Chittesh14
Part c please. Can you answer it and show me how you got to the answers thanks :smile:


My working out:


Posted from TSR Mobile


No images popping up.
Original post by RDKGames
No images popping up.


I put them up after I post... Lol
Original post by Chittesh14
I put them up after I post... Lol


Your method is correct, you rounded too much. I did too lol.

I used exact values on my calc and got x to be 36.777... which is 36.8 to 3 d.p.
Original post by RDKGames
Your method is correct, you rounded too much. I did too lol.

I used exact values on my calc and got x to be 36.777... which is 36.8 to 3 d.p.


Oh right :smile: thanks. I don't know why they do this lol, I get pissed off. The answers in the book are exact - if you round none of the values.


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Original post by RDKGames
Your method is correct, you rounded too much. I did too lol.

I used exact values on my calc and got x to be 36.777... which is 36.8 to 3 d.p.


What about this one, apparently x = 4. If you get that answer, can you please show working out in a picture plz


Posted from TSR Mobile
(edited 7 years ago)
Original post by RDKGames
Your method is correct, you rounded too much. I did too lol.

I used exact values on my calc and got x to be 36.777... which is 36.8 to 3 d.p.
image.jpeg

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