Question

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).

Answer 1

:)

Answer 2

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

Answer 3

2.645, not 2.65

Answer 4

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).

Answer 5

ok, so then I just sub the 2 angs form 180 and....
right?

Answer 6

First find one of the other angles.

Answer 7

ok I just did

Answer 8

79.2

Answer 9

something is not right

Answer 10

I think you found the sum of 60 and one of the other ones, 19.1.

Answer 11

is 2.645 right for the third side?

Answer 12

\[\frac{a}{\sin \theta}=\frac{\sqrt{7} a}{\sin 60^o}\]

Answer 13

i found one angle to be 19.1

Answer 14

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.

Answer 15

The side opposite 60 degrees would then have length,\[\sqrt{7}a\]

Answer 16

Yes, you're right.

Answer 17

how?

Answer 18

oh, yeah, i get it

Answer 19

How? Cosine rule, like you used before (if you're asking how I get sqrt(7)a?).

Answer 20

Once the other angle's found, you're done.

Answer 21

ok, subing 19.1+60 from 180 gives 100.9

Answer 22

but

Answer 23

when i try to find the last angle using sine rule, it gives me 79.2

Answer 24

which is right?

Answer 25

Yeah, you're right...

Answer 26

i did something wrong then

Answer 27

Bizarre

Answer 28

:)

Answer 29

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

Answer 30

the ratio of sides 1 to 3, the sine rule, the cosine rule and the 60 angle

Answer 31

Does your question say "the length of *the* two adjacent sides" or "the length of two adjacent sides"?

Answer 32

the

Answer 33

whats the difference?

Answer 34

'the' restricts your choice

Answer 35

it says adjacent

Answer 36

but i still dont see what I did wrong :(

Answer 37

Omg, it just dawned on me - there are two possible solutions!

Answer 38

One set will have (60, 9.1 and the other) and (60, 79.2, other)

Answer 39

One assumption was missed - that there is only one solution.

Answer 40

I constructed a triangle on the description in GeoGebra and have (60,9.1,100.9) as one solution.

Answer 41

:) so there is just one solution?

Answer 42

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

Answer 43

in *one* rotation

Answer 44

so, is it then imposible to find both of the angles correctly?

Answer 45

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).

Answer 46

so, how do I answer the quesstion in a work book? which set of angles do I choose?

Answer 47

ok..I'm trying to figure out a way to do it so that it doesn't take a millennium to type.

Answer 48

wouldn't the other angle be 30 and the other 90?

Answer 49

since the other is 60?

Answer 50

\[\frac{\sin \alpha}{3a}=\frac{\sin 60}{\sqrt{7}a}\rightarrow \sin \alpha = \frac{3\sqrt{3}}{2\sqrt{7}}\]

Answer 51

Now

Answer 52

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

Answer 53

where \[\sin^{-1}\frac{3\sqrt{3}}{2\sqrt{7}}\approx 79.1^o\]

Answer 54

So the possible angles you get as solutions when considering this combination of sides is\[79.1^o, 100.9^o\]

Answer 55

Try taking the sine of both of them in your calculator.

Answer 56

So you have NO CHOICE but to accept two solutions for this first combination of sides.

Answer 57

ok
I get it

Answer 58

THANKS ALOT

Answer 59

hang on...

Answer 60

ok

Answer 61

just one question, is it a right triangle? ^_^

Answer 62

the question does not specify

Answer 63

because if so, then the other 2 angles are 90 and 30, otherwise , loki's answer is true :)

Answer 64

Wait wait wait...

Answer 65

:)

Answer 66

When you take the arc of sine here you'll get two solutions for each angle, which are algebraically correct.

Answer 67

lol, alright

Answer 68

So, calling those angles alpha and beta, you have\[\alpha \in \left\{ 79.1,100.9 \right\}\]and\[\beta \in \left\{ 19.1, 160.9 \right\}\]

Answer 69

BUT

Answer 70

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.

Answer 71

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?

Answer 72

then alpha is 100.9 and beta is 19.1 ?

Answer 73

There are four possible combinations, but only ONE combination works

Answer 74

Yes

Answer 75

and he strikes again~ lol

Answer 76

and i;m drunk - came back from a dinner

Answer 77

but you've answered it , weirdly in such a state ._. did you get it andy?

Answer 78

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.

Answer 79

yes

Answer 80

and, lokisan, you dont sound like drunk

Answer 81

maybe half drunk ~

Answer 82

\[\left\{ \alpha, \beta|\alpha + \beta +60^o=180^o , \alpha \in \left\{ 79.1,100.9 \right\},\beta \in \left\{ 19.1,160.9 \right\} \right\}\]

Answer 83

discrete mathematics ^^" ...

Answer 84

yes...so the above set is 19.1 and 100.9.

Answer 85

Phew

Answer 86

Good question.

Answer 87

LOL

Answer 88

Happy with that BMFan?

Answer 89

yeah, it is hard, but i am damn happy

Answer 90

awesome