How to find trigonometric values using triangles as fractions? How to find trigonometric values using triangles? I only know how to find values of $30^\circ, 60^\circ, 45^\circ$  using triangle ( in fractional form ). Is there is any way to find values of angle $40^\circ, 90^\circ, 120^\circ, 180^\circ, 360^\circ$ as fractions
My calculator give me answer in decimals but i need answer in fractional form. Is there is any way to find out values using triangle like we find for $30^\circ, 60^\circ, 45^\circ$?
 A: By far the most interesting of your questions is the one about $40^\circ$, and I’ll leave that to the end. The trig functions of the other angles are determined by using the unit circle, as your trigonometry course should have explained to you. Take the circle $x^2+y^2=1$ in the plane, and, starting at the positive $x$-axis, make an angle $\theta$ with vertex at the origin, turning counterclockwise (anticlockwise if you’re British). Then the other leg of the angle will intersect the unit circle at a point $(\xi,\eta)$, and you have: $\sin\theta=\eta$, $\cos\theta=\xi$, and $\tan\theta=\eta/\xi$. This will allow you to answer your question for all your angles but $40^\circ$.
Now for the interesting question. For some angles, like $15^\circ$, you can get an expression for the trigonometric functions involving square roots, perhaps repeated. For others, such as $10^\circ$, $20^\circ$, and $40^\circ$, the cosine of the angle satisfies a polynomial with rational coefficients that is irreducible and cubic. The verification of this fact is an essential part of at least one of the proofs that there are angles that can’t be trisected with the traditional methods of compass and unmarked straightedge. In particular, you’re not going to be able to construct a triangle or describe in any decent geometric way a triangle whose angles include one of $40^\circ$. Even if you used the Cubic Formula, you wouldn’t get anywhere in this task. So your hope of finding a triangular description for $\sin40^\circ$ must remain forlorn.
