The length of the adjacent, opposite, and the hypotenuse of a triangle without using sOHcAHtOA

[myxkeyisstuckxxxx]

We know that:
x = a = sin(θ) = y/z
y = b = cos(θ) = x/z
z = c = tan(θ) = y/x

This is very well known and it is sometimes simplified by the word sOHcAHtOA.
Now, how do we find the value of these variables using an alternative method? Well, we can use the following formulas:
x = sqrt(z^2 - y^2)
y = sqrt(z^2 - x^2)
z = sqrt(x^2 + y^2)

Here is an example:
A triangle with its hypotenuse measuring 10 units, its opposite measuring 4 units, and its adjacent measuring x units. What the value of x?
The variables are equal to:
y = 4
z = 10

The value of x is equal to:
x = sqrt(z^2 - y^2)
x = sqrt(10^2 - 4^2)
x = sqrt(84)
x = 9

We can verify it by doing:
z = sqrt(x^2 + y^2)
z = sqrt(9^2 + 4^2)
z = sqrt(97)
z = 10

And then for the value of y

The value of x is equal to:
y = sqrt(z^2 - x^2)
y = sqrt(10^2 - 9^2)
y = sqrt(19)
y = 4

We can verify it by doing:
z = sqrt(x^2 + y^2)
z = sqrt(9^2 + 4^2)
z = sqrt(97)
z = 10

And lastly, for the value of z, which is simply just the usual Pythagorean Theorem we all know and love.

Is there an existing method similar to this?
And, is this correct?

 

[simpleauthority]

Looks mostly right, however sqrt(84) is not 9 and sqrt(97) is not 10 and sqrt(19) is not 4. All of this is actually the Pythagorean theorem arranged in different ways. There are also the Law of Cosines and Law of Sines respectively as well as some other fun trigonometry tricks.

I just took some notes going over a few things. Here they are:

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edit: the c = asin(c)/sin(a) should be the last page, not the second page. If it's the second page, just ignore it and go back at the end. Sorry.

 

[myxkeyisstuckxxxx]

Looks mostly right, however sqrt(84) is not 9 and sqrt(97) is not 10 and sqrt(19) is not 4. All of this is actually the Pythagorean theorem arranged in different ways. There are also the Law of Cosines and Law of Sines respectively as well as some other fun trigonometry tricks.

I just took some notes going over a few things. Here they are:

To view this content we will need your consent to set third party cookies.
For more detailed information, see our cookies page.

Accept third party cookies

edit: the c = asin(c)/sin(a) should be the last page, not the second page. If it's the second page, just ignore it and go back at the end. Sorry.

Yeah, its not exactly a whole number. sqrt(84) = 9.16515139, sqrt(97) = 9.8488578018, and sqrt(19) = 4.35889894354. I just rounded it since those are small deviation.