anonymous
  • anonymous
One corner of a triangle has a 60° angle and the length of the two adjacent sides are in ratio 1 : 3. Calculate the angles of the other triangle corners (0,1°:s precision, 1 point / correct angle).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
:)
anonymous
  • anonymous
I think I have to first find the size of the opposite side to 60 degree angle right? i used cosine rule to do so and found 2.65, however I dont know what to do now
anonymous
  • anonymous
2.645, not 2.65

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anonymous
  • anonymous
Yes, you're on the right track. Now use sine rule to find your angles...but, note you only have to use it once because the sum of the angles of a plane triangle is 180 degrees (and by the time you apply the sine rule, you'll have two of them).
anonymous
  • anonymous
ok, so then I just sub the 2 angs form 180 and.... right?
anonymous
  • anonymous
First find one of the other angles.
anonymous
  • anonymous
ok I just did
anonymous
  • anonymous
79.2
anonymous
  • anonymous
something is not right
anonymous
  • anonymous
I think you found the sum of 60 and one of the other ones, 19.1.
anonymous
  • anonymous
is 2.645 right for the third side?
anonymous
  • anonymous
\[\frac{a}{\sin \theta}=\frac{\sqrt{7} a}{\sin 60^o}\]
anonymous
  • anonymous
i found one angle to be 19.1
anonymous
  • anonymous
Your third side is right, assuming, in your ratio of 1:3, the side with the '1' has unit length. A more general assumption is that this side has length 'a', so that the other side has length 3a.
anonymous
  • anonymous
The side opposite 60 degrees would then have length,\[\sqrt{7}a\]
anonymous
  • anonymous
Yes, you're right.
anonymous
  • anonymous
how?
anonymous
  • anonymous
oh, yeah, i get it
anonymous
  • anonymous
How? Cosine rule, like you used before (if you're asking how I get sqrt(7)a?).
anonymous
  • anonymous
Once the other angle's found, you're done.
anonymous
  • anonymous
ok, subing 19.1+60 from 180 gives 100.9
anonymous
  • anonymous
but
anonymous
  • anonymous
when i try to find the last angle using sine rule, it gives me 79.2
anonymous
  • anonymous
which is right?
anonymous
  • anonymous
Yeah, you're right...
anonymous
  • anonymous
i did something wrong then
anonymous
  • anonymous
Bizarre
anonymous
  • anonymous
:)
anonymous
  • anonymous
In mathematics, if you end up with a contradiction, it's because one of your assumptions is wrong...so what assumption(s) were made?
anonymous
  • anonymous
the ratio of sides 1 to 3, the sine rule, the cosine rule and the 60 angle
anonymous
  • anonymous
Does your question say "the length of *the* two adjacent sides" or "the length of two adjacent sides"?
anonymous
  • anonymous
the
anonymous
  • anonymous
whats the difference?
anonymous
  • anonymous
'the' restricts your choice
anonymous
  • anonymous
it says adjacent
anonymous
  • anonymous
but i still dont see what I did wrong :(
anonymous
  • anonymous
Omg, it just dawned on me - there are two possible solutions!
anonymous
  • anonymous
One set will have (60, 9.1 and the other) and (60, 79.2, other)
anonymous
  • anonymous
One assumption was missed - that there is only one solution.
anonymous
  • anonymous
I constructed a triangle on the description in GeoGebra and have (60,9.1,100.9) as one solution.
anonymous
  • anonymous
:) so there is just one solution?
anonymous
  • anonymous
No, it's coming about because the arc of sine (in on rotation) has TWO angles whose sine is positive and the same value.
anonymous
  • anonymous
in *one* rotation
anonymous
  • anonymous
so, is it then imposible to find both of the angles correctly?
anonymous
  • anonymous
No, it's just that you collect both possible solutions from the arc of sine on each angle, and then put them in the appropriate combinations (i.e. so they add to 180).
anonymous
  • anonymous
so, how do I answer the quesstion in a work book? which set of angles do I choose?
anonymous
  • anonymous
ok..I'm trying to figure out a way to do it so that it doesn't take a millennium to type.
anonymous
  • anonymous
wouldn't the other angle be 30 and the other 90?
anonymous
  • anonymous
since the other is 60?
anonymous
  • anonymous
\[\frac{\sin \alpha}{3a}=\frac{\sin 60}{\sqrt{7}a}\rightarrow \sin \alpha = \frac{3\sqrt{3}}{2\sqrt{7}}\]
anonymous
  • anonymous
Now
anonymous
  • anonymous
in the arc of 360 degrees, \[\alpha = \sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\] degrees AND\[\alpha = 180- \sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\]
anonymous
  • anonymous
where \[\sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\approx 79.1^o\]
anonymous
  • anonymous
So the possible angles you get as solutions when considering this combination of sides is\[79.1^o, 100.9^o\]
anonymous
  • anonymous
Try taking the sine of both of them in your calculator.
anonymous
  • anonymous
So you have NO CHOICE but to accept two solutions for this first combination of sides.
anonymous
  • anonymous
ok I get it
anonymous
  • anonymous
THANKS ALOT
anonymous
  • anonymous
hang on...
anonymous
  • anonymous
ok
anonymous
  • anonymous
just one question, is it a right triangle? ^_^
anonymous
  • anonymous
the question does not specify
anonymous
  • anonymous
because if so, then the other 2 angles are 90 and 30, otherwise , loki's answer is true :)
anonymous
  • anonymous
Wait wait wait...
anonymous
  • anonymous
:)
anonymous
  • anonymous
When you take the arc of sine here you'll get two solutions for each angle, which are algebraically correct.
anonymous
  • anonymous
lol, alright
anonymous
  • anonymous
So, calling those angles alpha and beta, you have\[\alpha \in \left\{ 79.1,100.9 \right\}\]and\[\beta \in \left\{ 19.1, 160.9 \right\}\]
anonymous
  • anonymous
BUT
anonymous
  • anonymous
only certain combinations of those angles will give you a true conclusion here, since you have an additional constraint: that the angles\[\alpha, \beta, 60^o\]must sum to 180 degrees.
anonymous
  • anonymous
So you have to find those combinations elements from the set of alpha and beta that will allow you to get 180 (after you add 60 to them). You see?
anonymous
  • anonymous
then alpha is 100.9 and beta is 19.1 ?
anonymous
  • anonymous
There are four possible combinations, but only ONE combination works
anonymous
  • anonymous
Yes
anonymous
  • anonymous
and he strikes again~ lol
anonymous
  • anonymous
and i;m drunk - came back from a dinner
anonymous
  • anonymous
but you've answered it , weirdly in such a state ._. did you get it andy?
anonymous
  • anonymous
It's similar to a situation when you have to solve the quadratic equation, which might have something to do with length, and you get two solutions - one positive, one negative. You apply an additional constraint (i.e. physical measurements aren't negative) and discard one of the solutions. Here, the constraint is that you can only take those angles whose sum will be 180.
anonymous
  • anonymous
yes
anonymous
  • anonymous
and, lokisan, you dont sound like drunk
anonymous
  • anonymous
maybe half drunk ~
anonymous
  • anonymous
\[\left\{ \alpha, \beta|\alpha + \beta +60^o=180^o , \alpha \in \left\{ 79.1,100.9 \right\},\beta \in \left\{ 19.1,160.9 \right\} \right\}\]
anonymous
  • anonymous
discrete mathematics ^^" ...
anonymous
  • anonymous
yes...so the above set is 19.1 and 100.9.
anonymous
  • anonymous
Phew
anonymous
  • anonymous
Good question.
anonymous
  • anonymous
LOL
anonymous
  • anonymous
Happy with that BMFan?
anonymous
  • anonymous
yeah, it is hard, but i am damn happy
anonymous
  • anonymous
awesome

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