Understanding finding sine of radian angle So given the problem to find the sine of angle $7\pi / 6$ without using a calculator.
I reasoned that a half unit circle has radian measure of $\pi$. So we go one half circle
and another $1/6$ which leads us into the 3rd quadrant. So the value will be negative.
I then drew a right triangle with the hypotenuse equal to the radius of the unit circle, which is 1.

Now, since I know that the angle $O$ in the right triangle is 30 degree, and we are in a $30-60-90$ triangle, I concluded the sine I am looking for is $-1/2$.
But assuming I wouldn’t know sine of $30$ deg or I wouldn’t want to rely on the right triangle definition of sine - how could I find out the sine of $7\pi / 6$?
 A: $\sin\left(\dfrac{7\pi}{6}\right) = \sin\left(\dfrac{\pi}{6} + \pi\right) = -\sin\left(\dfrac{\pi}{6}\right) = -\dfrac{1}{2}$
Try to convince yourself of the identity $\sin(x+\pi) = -\sin(x)$ (hint: just draw the unit circle and see where sine is positive and negative). You can find the value of $\sin\left(\dfrac{\pi}{6}\right)$ by drawing an equilateral triangle and splitting it in two (try it, it's something everyone should do once in their life). By the way, there's nothing wrong with your reasoning, it's completely correct.
A: TL;DR: What you did was completely correct and does not rely on a right-triangle definition of the sine function.

You can use triangles to help you find values of the sine function while still using the unit circle to define the sine function.
For the angle $\frac{7\pi}{6},$ you have already shown how to use the unit-circle
definition to demonstrate that $\sin\left(\frac{7\pi}{6}\right)$ is the $y$-coordinate of the end of a radial segment of the unit circle, making an angle $\frac\pi6$ below the negative $y$-axis.
As you observed, if you drop a perpendicular from the end of that radial segment to the $y$-axis, you form a $30$-$60$-$90$-degree right triangle.
You are not using that triangle to define the sine, but you happen to know that the leg opposite the $30$-degree angle is half the length of the hypotenuse, and since the hypotenuse is the radial segment, already known to have length $1,$ the length of the opposite segment is $\frac12.$
That is, the end of the radius is $\frac12$ unit directly below the $y$-axis, so it has $y$-coordinate $-\frac12,$ and therefore,
according to the unit-circle definition of the sine function,
$$\sin\left(\frac{7\pi}{6}\right) = -\frac12.$$
If you did not already know the ratios of the sides of a $30$-$60$-$90$-degree right triangle, then (as observed already in another answer) you could derive them by cutting an equilateral triangle in half.
Most angles do not have a geometric construction that gives us exact values of their trigonometric functions, but it happens that the angle $\frac\pi6$ does have such a construction and it gives a particularly simple result, so we have schoolchildren memorize that result soon after introducing the idea of measuring the angles of a triangle.
Indeed, for almost any other angle there is not such a simple way to find the sine;
you can construct a triangle within the unit circle whose legs are the same magnitude as the angle's sine and cosine, but you will not be able to write the exact values of that sine and cosine as simple ratios of integers. You cannot even write those values in any finite expression involving integers, addition, subtraction, multiplication, and division.
That is why we usually settle for an approximate value expressed as a decimal fraction.
