How to Convert Tangents & Degrees

Video Transcript

"Here's the basic info on how to use tangents. In a right triangle, one with ninety degrees, as one of its angles, you can always find the tangent of one of the other angles. Here I'm going to call the other angle theta. Tangent of theta is defined to be the ratio, or the fraction, of the two lengths. The opposite lengths of the side from theta, and the length of the adjacent side to theta. That's the one that theta shares with the right angle. For an example, if my right triangle has three sides of lengths three, four, and five, and this angle theta is shown, then tangent of theta uses the three and the four. It's the ratio three fourths. For an angle of general size, anywhere from negative infinity to infinity, we can describe tangent of theta by looking at the unit circle. This circle has a radius one. So it crosses the X and Y axes at one. Here's the angle theta drawn on the unit circle. And you can tell by looking at this angle theta, it's the same as a picture of a right triangle. If I either draw the vertical line here, or here. Those are two similar triangles, so they'll have the same tangent of theta, the same theta. Now, however, I can describe the opposite and the adjacent sides by using the X and Y coordinates. I can use the X and Y coordinates of the point where that angle, the ray making that angle, crosses the circle. It's a tangent theta, equals Y divided by X. And here's what happens if I decide to use the bigger triangle. Now, the adjacent side is the radius of the circle of length one. So when I divide by one, I get exactly what I started with. In other words, tangent of theta is precisely the length of this vertical line. That's the line which is tangent to the circle, hence the name tangent. Now, in a triangle, where theta is the angle, if you happen to know the tangent theta, we'll call it A, then you can work backwards to find out what the angle is. Theta, the angle, is equal to the arc tangent of A, or sometimes it's written with the tangent inverse notation. This will always give you a result in a real triangle between zero and pie over two, or ninety degrees. In general, if you give any argument to inverse tangent, the result is defined to be in between negative pie over two, and pie over two. And here's what the graph of invert tangent looks like."