Right triangle

[Probability]

I'm not a mathematician I'm more of a leaner and a hobby type person, sowith that said please bear with me on this subject...

I'm looking at a right angled triangle and the only angle I know is 90 degrees. The two remaining angles could be anything that when added together plus the 90 equal 180. That much is striaght forward.

My question...

If I have a right angle triangle not knowing the side lengths and only the 90 degree angle, is it possible mathematically to work out the two remaining angles?

Also, once the two remaining angles are known, then to calculate the lengths of the sides, can SOH CAH TOA solve all three sides?

Using sine theta and tan theta I've managed to work out the angles but from the angles and above, and from a scale drawing, I don't seem to be able to calculate the length of the sides and get them all correct!

Any guidance very much appreciated.

 

[mario99]

If I have a right angle triangle not knowing the side lengths and only the 90 degree angle, is it possible mathematically to work out the two remaining angles?

We want you to figure out the answer by looking at one of the trigonometric functions, say, sine:

$\displaystyle \sin \theta = \frac{a}{h}$

where $a$ is one of the small sides of the triangle and $h$ is the hypotenuse.

How many unknowns do you have there?

 

[Dr.Peterson]

If I have a right angle triangle not knowing the side lengths and only the 90 degree angle, is it possible mathematically to work out the two remaining angles?

Another way to think about is to imagine trying to draw this triangle, knowing no side lengths and neither of the acute angles. How would you do it? Could you be sure you had drawn the correct triangle?

 

[lev888]

The two remaining angles could be anything that when added together plus the 90 equal 180. That much is striaght forward.

You may want to reread the above.

 

[Probability]

We want you to figure out the answer by looking at one of the trigonometric functions, say, sine:

$\displaystyle \sin \theta = \frac{a}{h}$

where $a$ is one of the small sides of the triangle and $h$ is the hypotenuse.

How many unknowns do you have there?

I'm trying to look at this problem from first principles. We can all draw a right triangle and measure the side lengths and then divide those lengths to show the angle, but what I'm trying to establish here is the other way of working that problem out.

In my right triangle I have two angles unknown and three side lengths unknown. Without a scale drawing first to measure the lengths of the sides to calculate the angles, I'm not convinced its possible to mathematically work it out?

 

[Probability]

Another way to think about is to imagine trying to draw this triangle, knowing no side lengths and neither of the acute angles. How would you do it? Could you be sure you had drawn the correct triangle?

I think a right triangle is a triangle with one 90 degree angle and two acute angles. I have no doubt with that but I don't think its possible to calculate the acute angles without knowing any of the side lengths, which is what I'm trying to establish.

 

[lev888]

We can all draw a right triangle and measure the side lengths and then divide those lengths to show the angle, but what I'm trying to establish here is the other way of working that problem out.

What problem? Please explain clearly what that problem is. What are the givens and what do you need to find?
Original problem: Given an arbitrary right triangle you drew, measure sides, calculate angles. What is the "the other way" problem? Shouldn't it be "pick some triangle dimensions sufficient to define it, then draw it"? And knowing just that the triangle has one 90 degree angle is obviously not enough to define it.

 

[mario99]

I'm trying to look at this problem from first principles. We can all draw a right triangle and measure the side lengths and then divide those lengths to show the angle, but what I'm trying to establish here is the other way of working that problem out.

In my right triangle I have two angles unknown and three side lengths unknown. Without a scale drawing first to measure the lengths of the sides to calculate the angles, I'm not convinced its possible to mathematically work it out?

If the length of each side is unknown, it is possible to find the angles if you know the relationship between the ratios of the sides of the triangle.

For example, if the ratio of the hypotenuse to the leg $a$ is $h:a = 3:1$, yes it is possible to find the angles. If everything is unknown, the two angles can be anything which means we have $\infty$ possibility. Reread post #4 for lev888.

 

[Otis]

once the two remaining angles are known, then to calculate the lengths of the sides, can SOH CAH TOA solve all three sides?

Hi. Here's a related question to consider. If I enlarge or reduce a triangle (like using a photocopier), do the angle measures change?

 

[Dr.Peterson]

In my right triangle I have two angles unknown and three side lengths unknown. Without a scale drawing first to measure the lengths of the sides to calculate the angles, I'm not convinced its possible to mathematically work it out?

I think a right triangle is a triangle with one 90 degree angle and two acute angles. I have no doubt with that but I don't think its possible to calculate the acute angles without knowing any of the side lengths, which is what I'm trying to establish.

You're right: It can't be done, unless there is something you do know, besides the one right angle. That's what we're all leading you toward, but you seem to be already there. You just need to take the step from "not convinced it's possible" to "convinced it's not possible".

Or, of course, if there is anything else you do know, such as ratios of sides (or of angles), tell us.

 

[MathNugget]

I think a right triangle is a triangle with one 90 degree angle and two acute angles. I have no doubt with that but I don't think its possible to calculate the acute angles without knowing any of the side lengths, which is what I'm trying to establish.

Would you settle for a reductio ad absurdum?
Suppose there was a way to find the angles of the right triangle without knowing anything about it besides having a 90 angle. Wouldn't this mean all right triangles have the same angles? Therefore, finding 2 right triangles with different angles would give us a contradiction, so the supposition would be false; you cannot find the angles of a right triangle without knowing any of the side lengths or 1 of the acute angles.

Maybe I am misunderstanding it... But if you talk about uniquely determining a triangle ( so not saying the angles are x and 90-x), I think the concept above is simplest available to satisfy your need for a mathematical proof of your 'theorem'. As far as I know, proving something is impossible is done with the technique above; we suppose it is possible, and prove it is not.

 

[Probability]

I'll return as soon as I've finished my two right triangle drawings and maths workout. I'll give a full explanation in the next post probably tomorrow evening. Thanks for all replies so far. Please be patient with me I'm not a mathematician, just an hobbyist and leaner.

 

[MathNugget]

I'll return as soon as I've finished my two right triangle drawings and maths workout. I'll give a full explanation in the next post probably tomorrow evening. Thanks for all replies so far. Please be patient with me I'm not a mathematician, just an hobbyist and leaner.

Can't speak for everyone else, but in this day and age, I'm surprised to see anyone interested in math (of any level). And being interested in complete proofs, on top of that. Good luck with the project.
I'll try to check it out, got me curious about where this exercise is going.