How could I find a side of a right triangle that gives me the length of the hypotenuse and two of the angles, but doesn't give me sides a or b? (will give medal & fan)

Question

anonymous · Answer

Specifically, the right triangle i'm working with has a hypotenuse of 25, and two of the angles are 90 degrees (obviously) and 55 degrees... which would make the other angle 35 degrees.

anonymous · Answer

But I need to find side b of the triangle.

anonymous · Answer

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anonymous · Answer

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whpalmer4 · Answer

Do you know about the trigonometric ratios?

anonymous · Answer

Not exactly... I might, but I need a refresher.

whpalmer4 · Answer

SOH CAH TOA

Sin of an angle = Opposite / Hypotenuse
Cos of an angle = Adjacent / Hypotenuse
Tan of an angle = Opposite / Adjacent

We know the angles, and we know the hypotenuse

whpalmer4 · Answer

So we can say, for example, that $$\sin 35 ^\circ = \frac{a}{25}$$

anonymous · Answer

Oh okay, I see what you did there.

whpalmer4 · Answer

Then $$25\sin 35^\circ = a$$

anonymous · Answer

so a = 14.34 (rounded to two decimal places)?

whpalmer4 · Answer

Indeed!

whpalmer4 · Answer

then you could find the other one with Pythagorean theorem or one of the other trig ratios

anonymous · Answer

Okay so when all is said and done b = 2.04 (rounded)?

anonymous · Answer

*3.04

anonymous · Answer

b + 14.34^2 = 25^2
b + 205.6356 = 625
b = 625/205.6356
b = 3.04

whpalmer4 · Answer

No, that's not quite right...

You are using a little-known variation of the Pythagorean theorem there:
$$a^2+b = c^2$$:-)

whpalmer4 · Answer

As a reasonableness check, look at your triangle:  55 and 35 degrees aren't quite equal, but they are both pretty close to 45 degrees, which would give you an isosceles triangle with a = b....you've got something more like a = 5b

anonymous · Answer

so wait, a = 14.34 but wouldn't b be larger than that?

anonymous · Answer

I'm confused now lol

whpalmer4 · Answer

yes, $b$ is larger than $a$.  If we had a 45/45/90 triangle, a would equal b.  I'm saying the angles are close enough to equal that the sides should be much closer to each other in length.  A triangle with the sides you computed is impossible — think about it!  The other two sides added together must be at least the length of the hypotenuse, right? :-)

anonymous · Answer

So then... a^2 + b^2 = c^2
14.34^2 + b^2 = 25^2

Am I doing this right so far?

whpalmer4 · Answer

yes!

whpalmer4 · Answer

before you did$$14.34^2 + b = 25^2$$

anonymous · Answer

okay so
205.6356 + b^2 = 625

b^2 = 625/205.6356
b^2 = 3.04
b = 1.74

... still not working the way i want it to...

anonymous · Answer

ohhh wait i get it
i'm dividing instead of subtracting

anonymous · Answer

b = 20.48

whpalmer4 · Answer

Yes, that's the right answer!

anonymous · Answer

yay! thank you!