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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).

  • 3 years ago
  • 3 years ago

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  1. BecomeMyFan=D
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    :)

    • 3 years ago
  2. BecomeMyFan=D
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    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

    • 3 years ago
  3. BecomeMyFan=D
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    2.645, not 2.65

    • 3 years ago
  4. lokisan
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    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).

    • 3 years ago
  5. BecomeMyFan=D
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    ok, so then I just sub the 2 angs form 180 and.... right?

    • 3 years ago
  6. lokisan
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    First find one of the other angles.

    • 3 years ago
  7. BecomeMyFan=D
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    ok I just did

    • 3 years ago
  8. BecomeMyFan=D
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    79.2

    • 3 years ago
  9. BecomeMyFan=D
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    something is not right

    • 3 years ago
  10. lokisan
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    I think you found the sum of 60 and one of the other ones, 19.1.

    • 3 years ago
  11. BecomeMyFan=D
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    is 2.645 right for the third side?

    • 3 years ago
  12. lokisan
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    \[\frac{a}{\sin \theta}=\frac{\sqrt{7} a}{\sin 60^o}\]

    • 3 years ago
  13. BecomeMyFan=D
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    i found one angle to be 19.1

    • 3 years ago
  14. lokisan
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    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.

    • 3 years ago
  15. lokisan
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    The side opposite 60 degrees would then have length,\[\sqrt{7}a\]

    • 3 years ago
  16. lokisan
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    Yes, you're right.

    • 3 years ago
  17. BecomeMyFan=D
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    how?

    • 3 years ago
  18. BecomeMyFan=D
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    oh, yeah, i get it

    • 3 years ago
  19. lokisan
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    How? Cosine rule, like you used before (if you're asking how I get sqrt(7)a?).

    • 3 years ago
  20. lokisan
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    Once the other angle's found, you're done.

    • 3 years ago
  21. BecomeMyFan=D
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    ok, subing 19.1+60 from 180 gives 100.9

    • 3 years ago
  22. BecomeMyFan=D
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    but

    • 3 years ago
  23. BecomeMyFan=D
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    when i try to find the last angle using sine rule, it gives me 79.2

    • 3 years ago
  24. BecomeMyFan=D
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    which is right?

    • 3 years ago
  25. lokisan
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    Yeah, you're right...

    • 3 years ago
  26. BecomeMyFan=D
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    i did something wrong then

    • 3 years ago
  27. lokisan
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    Bizarre

    • 3 years ago
  28. BecomeMyFan=D
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    :)

    • 3 years ago
  29. lokisan
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    In mathematics, if you end up with a contradiction, it's because one of your assumptions is wrong...so what assumption(s) were made?

    • 3 years ago
  30. BecomeMyFan=D
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    the ratio of sides 1 to 3, the sine rule, the cosine rule and the 60 angle

    • 3 years ago
  31. lokisan
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    Does your question say "the length of *the* two adjacent sides" or "the length of two adjacent sides"?

    • 3 years ago
  32. BecomeMyFan=D
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    the

    • 3 years ago
  33. BecomeMyFan=D
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    whats the difference?

    • 3 years ago
  34. lokisan
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    'the' restricts your choice

    • 3 years ago
  35. BecomeMyFan=D
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    it says adjacent

    • 3 years ago
  36. BecomeMyFan=D
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    but i still dont see what I did wrong :(

    • 3 years ago
  37. lokisan
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    Omg, it just dawned on me - there are two possible solutions!

    • 3 years ago
  38. lokisan
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    One set will have (60, 9.1 and the other) and (60, 79.2, other)

    • 3 years ago
  39. lokisan
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    One assumption was missed - that there is only one solution.

    • 3 years ago
  40. lokisan
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    I constructed a triangle on the description in GeoGebra and have (60,9.1,100.9) as one solution.

    • 3 years ago
  41. BecomeMyFan=D
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    :) so there is just one solution?

    • 3 years ago
  42. lokisan
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    No, it's coming about because the arc of sine (in on rotation) has TWO angles whose sine is positive and the same value.

    • 3 years ago
  43. lokisan
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    in *one* rotation

    • 3 years ago
  44. BecomeMyFan=D
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    so, is it then imposible to find both of the angles correctly?

    • 3 years ago
  45. lokisan
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    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).

    • 3 years ago
  46. BecomeMyFan=D
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    so, how do I answer the quesstion in a work book? which set of angles do I choose?

    • 3 years ago
  47. lokisan
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    ok..I'm trying to figure out a way to do it so that it doesn't take a millennium to type.

    • 3 years ago
  48. sstarica
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    wouldn't the other angle be 30 and the other 90?

    • 3 years ago
  49. sstarica
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    since the other is 60?

    • 3 years ago
  50. lokisan
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    \[\frac{\sin \alpha}{3a}=\frac{\sin 60}{\sqrt{7}a}\rightarrow \sin \alpha = \frac{3\sqrt{3}}{2\sqrt{7}}\]

    • 3 years ago
  51. lokisan
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    Now

    • 3 years ago
  52. lokisan
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    in the arc of 360 degrees, \[\alpha = \sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\] degrees AND\[\alpha = 180- \sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\]

    • 3 years ago
  53. lokisan
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    where \[\sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\approx 79.1^o\]

    • 3 years ago
  54. lokisan
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    So the possible angles you get as solutions when considering this combination of sides is\[79.1^o, 100.9^o\]

    • 3 years ago
  55. lokisan
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    Try taking the sine of both of them in your calculator.

    • 3 years ago
  56. lokisan
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    So you have NO CHOICE but to accept two solutions for this first combination of sides.

    • 3 years ago
  57. BecomeMyFan=D
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    ok I get it

    • 3 years ago
  58. BecomeMyFan=D
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    THANKS ALOT

    • 3 years ago
  59. lokisan
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    hang on...

    • 3 years ago
  60. BecomeMyFan=D
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    ok

    • 3 years ago
  61. sstarica
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    just one question, is it a right triangle? ^_^

    • 3 years ago
  62. BecomeMyFan=D
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    the question does not specify

    • 3 years ago
  63. sstarica
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    because if so, then the other 2 angles are 90 and 30, otherwise , loki's answer is true :)

    • 3 years ago
  64. lokisan
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    Wait wait wait...

    • 3 years ago
  65. BecomeMyFan=D
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    :)

    • 3 years ago
  66. lokisan
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    When you take the arc of sine here you'll get two solutions for each angle, which are algebraically correct.

    • 3 years ago
  67. sstarica
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    lol, alright

    • 3 years ago
  68. lokisan
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    So, calling those angles alpha and beta, you have\[\alpha \in \left\{ 79.1,100.9 \right\}\]and\[\beta \in \left\{ 19.1, 160.9 \right\}\]

    • 3 years ago
  69. lokisan
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    BUT

    • 3 years ago
  70. lokisan
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    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.

    • 3 years ago
  71. lokisan
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    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?

    • 3 years ago
  72. sstarica
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    then alpha is 100.9 and beta is 19.1 ?

    • 3 years ago
  73. lokisan
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    There are four possible combinations, but only ONE combination works

    • 3 years ago
  74. lokisan
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    Yes

    • 3 years ago
  75. sstarica
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    and he strikes again~ lol

    • 3 years ago
  76. lokisan
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    and i;m drunk - came back from a dinner

    • 3 years ago
  77. sstarica
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    but you've answered it , weirdly in such a state ._. did you get it andy?

    • 3 years ago
  78. lokisan
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    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.

    • 3 years ago
  79. BecomeMyFan=D
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    yes

    • 3 years ago
  80. BecomeMyFan=D
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    and, lokisan, you dont sound like drunk

    • 3 years ago
  81. sstarica
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    maybe half drunk ~

    • 3 years ago
  82. lokisan
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    \[\left\{ \alpha, \beta|\alpha + \beta +60^o=180^o , \alpha \in \left\{ 79.1,100.9 \right\},\beta \in \left\{ 19.1,160.9 \right\} \right\}\]

    • 3 years ago
  83. sstarica
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    discrete mathematics ^^" ...

    • 3 years ago
  84. lokisan
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    yes...so the above set is 19.1 and 100.9.

    • 3 years ago
  85. lokisan
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    Phew

    • 3 years ago
  86. lokisan
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    Good question.

    • 3 years ago
  87. sstarica
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    LOL

    • 3 years ago
  88. lokisan
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    Happy with that BMFan?

    • 3 years ago
  89. BecomeMyFan=D
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    yeah, it is hard, but i am damn happy

    • 3 years ago
  90. lokisan
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    awesome

    • 3 years ago
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