# Finding angles of right triangle WITHOUT calculator

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A couple of days ago in school we had a test, in one of the tasks we were given a right triangle. It were painted on squared paper(is this the right word?) so I could figure out that the site lengths was this: a = sqrt(3*3+3*3) = sqrt(18) = ~4.2 b = sqrt(4*4+4*4) = sqrt(32) = ~5.7 c = sqrt(5*5+1*1) = sqrt(50) = ~7.1 The first task was to find the longest site which is c(sqrt(50)), but then the hard part came, we were to find the smallest angle(not which one it was, but how many degrees it was) the problem is that we were not allowed to bring calculators so I couldn't use these(sin is to hard to be calculated without calculator): A = asin(a/c) and B = asin(b/c) So does anyone have an idea how this was supposed to be solved? I'm pretty sure we shouldn't use sinous, because we are in 8th grade and our teacher have never mentioned sin, cos, tan etc. BTW I ended up just guessing(38 degrees) [Edited by - CTar on June 7, 2005 4:12:22 PM]

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Can you post the diagram? It may contain more information than what you provided.

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I don't have a scanner available, but I have reproduced it in Paint:

That is all the info we got

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Ah, that's the special right triangle often called the 3-4-5 triangle, because 3*3+4*4=5*5. The interior angles of that triangle were taught to me by rote.

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First:
c = sqrt(5*5+1*1) = ...

Should be:
c = sqrt(7*7+1*1) = ...

The top angle is 90 degrees.
The side to the left is 4 units and side to right is 3 unit (1 unit = sqrt(2)).

As Mastaba said it's a 3-4-5 triangle.
It is a right triangle with the sides 3 and 4.

Now, I guess you should know the angles in such triangle if you haven't started with sinus or equal methods.

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knowing that 2 sides is 3 and 4 units, you find that third side is 5.
Now, you need to find angle a that
sin(a)=3/5

I cheated a bit and computed it with calculator. Result:
36.8698976458440213 degrees
I do not think it can be obtained in nice way(using trig identities). You can do some approximations, though, or remember this value that seems useless.

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I don't think there is any solution that would let you find what the angles are exactly.

Either:
- Your prof. wanted an approximation (doable);
- You should have known by heart what are the angles of the 3-4-5 triangle (pointless);
- The expected answer was just arcsin(3/5). It's perfectly correct and precise, that's what I would have written down. You know, the exact digits that makes the number are not that important. More important is what the number *is*.

Have a nice day,
jods

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You can make a pretty good estimation though.

C = Pi * d = 7Pi

You have a triangle called c.

Side1 = 7
Side2 = 1

If you divide both with two the angles remains the same.

Side1 = 3.5 (the radius of the circle)
Side2 = 0.5

Here is the estimation part.
This angle moves about half a square along the circumference.

The relation is:
Side2 / C = answer / 360

answer = Side2 / C * 360 = 0.5 / 7Pi * 360 0 8.1851

The angle we are looking for is 45 - the small angle = 45 - 8.1851 = 36.815

Pretty close to the answer but...

EDIT: Actually d is sqrt(50) but that is about 7...

[Edited by - __fold on June 8, 2005 9:09:55 AM]

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Thanks for the help everyone, I guess next time I get such a question I'll just write arcsin(0,6) as the answer.

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We didnt do trigg till grade 11..

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