>>7794378
idk I think the angles there are just hard to work with and you can derive just about all the other angles with the 45-45-90 and 30-60-90 anyways? again idk

But in all textbooks it says that these are the *only* exact trigonometric ratios.

>>7794393
I think you're right: http://www.intmath.com/blog/mathematics/how-do-you-find-exact-values-for-the-sine-of-all-angles-6212

If you are right then there are a lot of misleading textbooks and websites out there...

My teach said you can find exact values for pretty much any angles, they just might be wonky numbers

>>7794378

With all my undergraduate and graduate hours of mathematics, I don't have a clue what you are getting at.

Can you precisely define what you are talking about?

>>7794467
OP here, I'll explain it in a clearer way.

Textbooks and a number of websites I have looked at claim that all trigonometric values (sin, cos and tan) of almost all angles are **approximations given to a certain number of decimal places**. The *only* exceptions are for the angles of triangles with sides of length 1, 1, sqrt(2); and 1, 2, sqrt(3) (the ones in the pic). But this seems plain wrong to me, because I can immediately think of right-angled a triangle with sides 3, 4, 5 that would yield an exact trigonometric value and there are probably many others.

The textbooks I refer to are highschool level.. Maybe they are oversimplifying (which they really shouldn't!).
Note: where sqrt(3) means the square root of 3.

>>7794502

I don't see how the sin(45) can be expressed as an exact value to any finiite number of decimal places.

>>7794502
Exact does not mean rational; [math]\sqrt{2}[/math] for example is exact but it isn't rational.

I'm guessing they're taking into account the inverse as well i.e. [math]\sin^{-1}{\theta},\cos^{-1}{\theta}[/math] are also exact.

But they're still wrong. You can still get exact values for any [math]\displaystyle\sin{\left(\frac{3n}{2^m}\right)°},\cos{\left(\frac{3n}{2^m}\right)°}[/math] where [math]m, n[/math] are positive integers eg.
3° and 87° [math]\displaystyle\left(2(1-\sqrt3)\sqrt{5+\sqrt5}+(\sqrt{10}-\sqrt2)(\sqrt3+1),2(1+\sqrt3)\sqrt{5+\sqrt5}+(\sqrt{10}-\sqrt2)(\sqrt3-1),16\right)[/math],
1.5° and 88.5° [math]\displaystyle\left(\left(\sqrt{2+\sqrt2}\right)\left(\sqrt{15}+\sqrt3-\sqrt{10-2\sqrt5}\right) - \left(\sqrt{2-\sqrt2}\right)\left(\sqrt{30-6\sqrt5}+\sqrt5+1\right),\left(\sqrt{2+\sqrt2}\right)\left(\sqrt{30-6\sqrt5}+\sqrt5+1\right) + \left(\sqrt{2-\sqrt2}\right)\left(\sqrt{15}+\sqrt3-\sqrt{10-2\sqrt5}\right),16\right)[/math].

(Hopefully LaTeX holds up this time; if not just copy and paste this whole post into the TeX preview.)

>>7794559
And lo and behold, it doesn't hold up.

>>7794561
put more white space around the commands
the latex here is a bit brain damaged:

[math] \displaystyle \sin{ \left( \frac{3n}{2^m} \right)°}, \cos{ \left( \frac{3n}{2^m} \right)°}[/math] where m,n are positive integers eg.
3° and 87° [math] \displaystyle \left(2(1-\sqrt3)\sqrt{5+ \sqrt5}+( \sqrt{10}- \sqrt2)( \sqrt3+1),2(1+ \sqrt3) \sqrt{5+ \sqrt5}+( \sqrt{10}- \sqrt2)( \sqrt3-1),16 \right)[/math],
1.5° and 88.5° [math] \displaystyle \left( \left( \sqrt{2+ \sqrt2} \right) \left( \sqrt{15}+ \sqrt3- \sqrt{10-2 \sqrt5} \right) - \left( \sqrt{2- \sqrt2} \right) \left( \sqrt{30-6 \sqrt5}+ \sqrt5+1 \right), \left( \sqrt{2+ \sqrt2} \right) \left( \sqrt{30-6 \sqrt5}+ \sqrt5+1 \right) + \left( \sqrt{2- \sqrt2} \right) \left( \sqrt{15}+ \sqrt3-\sqrt{10-2 \sqrt5} \right),16 \right)[/math]

Thanks for the help! No textbook should oversimplify at the expense of accurate true information but it does happen...

Can you do the same for degrees that are not integers? e.g. 32.23412354 or whatever degrees?

>>7794623
Only if the degrees are nice enough to fulfill the above conditions. You'll notice that 1.5 isn't an integer.