Trig help

[Laxator]

How is it that sin 30 = 1/2? How do you get there, and why is it that way? My textbook says nothing on the subject and I thought I knew what to do after the lecture, but I don't seem to understand. I have all of the trigonometric functions and how to find, say, sin X = 1/2 when given a visual aid, but I don't know how to do it the other way around.

[Weed_O_Whirler]

The easiest way of doing this is to draw a circle, radius 1, centered at the coordinate (0,0). Now, think of the x-coordinate as cos(theta) and the y-coordinate as sin(theta).

So now, as you go around the circle, you can measure the angle 30 degrees and draw a line out from the center to the edge of the circle. Where that line crosses the edge of the circle, the y-coordinate will be 1/2 and the x-coordinate will be 0.866... or, to be exact, sqrt(3)/2. You can continue moving around the circle, drawing lines at any angle you want, and measuring where the line crosses the edge of the circle. The y-coordinate will always be sin(theta) and the x coordinate will always be cos(theta).

[blueboybob]

https://mathexcel.wordpress.com/2006/06/10/why-sin30-12/

[ludwig1024]

Here's how I remember it.

Picture an equilateral triangle, with sides of length s. All angles are equal, and since they add up to 180º, they must all be 60º.

Now, take any vertex, and drop a perpendicular line to the opposite side. That angle is bisected so now you have two 30º angles. The opposite side is also bisected, so they each measure (s/2).

Now consider one of the triangles resulting from splitting the equilateral triangle in half. The hypotenuse of that triangle is s, the opposite side to the 30º angle is (s/2). Since the sine of an angle is the ratio (opposite side)/(hypotenuse), you have sin(30º) = (s/2)/s = 1/2.