1.) This is a 30º triangle.
As you can see, the properties of a 30º SRT are that the side opposite the angle has a value of x, the side adjacent to the angle has a value of x radical 3, and the hypotenuse has a value of 2x.
We simplify the hypotenuse to make it equal to 1, but whatever we do to one side must be done to the other sides. When we divide the hypotenuse by 2x, we also divide the opposite and adjacent sides by 2x, which reduce to 1/2 and radical 3 over 2 respectively.
When we place the triangle on a coordinate plane so that the angle is at the origin, we can figure out the coordinates of the points of the triangle. The point with the 30º angle is at (0,0), the point with the right angle is at (radical 3 over 2, 0), and the last point is at (radical 3 over 2, 1/2).
2.) This is a 45º triangle.
As you can see, the properties of a 45º SRT are that the hypotenuse has a value of x radical 2 and the other two sides have a value of x.
We simplify the hypotenuse to make it equal to 1, but whatever we do to one side must be done to the other sides. When we divide the hypotenuse by x radical 2, we also divide the other sides by x radical 2, which both reduce to radical 2 over two.
When we place the triangle on a coordinate plane so that the angle is at the origin, we can figure out the coordinates of the points of the triangle. The point with the 45º angle is at (0,0), the point with the right angle is at (radical 2 over 2, 0), and the last point is at (radical 2 over 2, radical 2 over 2).
3.) This is a 60º triangle.
As you can see, the properties of a 60º SRT are that the side opposite the angle has a value of x radical 3, the side adjacent to the angle has a value of x, and the hypotenuse has a value of 2x.
We simplify the hypotenuse to make it equal to 1, but whatever we do to one side must be done to the other sides. When we divide the hypotenuse by 2x, we also divide the opposite and adjacent sides by 2x, which reduce to radical 3 over 2 and 1/2 respectively.
When we place the triangle on a coordinate plane so that the angle is at the origin, we can figure out the coordinates of the points of the triangle. The point with the 60º angle is at (0,0), the point with the right angle is at (1/2, 0), and the last point is at (1/2, radical 3 over 2).
4.) This activity helps us derive the Unit Circle by showing us how to get the ordered pairs that make up the circle. When we make the hypotenuse equal to 1, we essentially set it as the radius of the circle. We also now understand which ordered pairs are connected with which SRTs.
5.) For this next section, we will be focusing specifically on the coordinates of the radius- or rather, how they change when the SRTs are moved to different quadrants.
When we move the 30º triangle to the second quadrant, the y value of the ordered pair stays the same, but the x value is now negative because it is on the negative side of the x axis.
When we move the 45º triangle to the second quadrant, the x value is still negative because it is on the negative side of the x axis, and the y value is now also negative because it is on the negative side of the y axis as well.
When we move the 60º triangle to the second quadrant, the x value is positive again because it is back on the positive side of the x axis, and the y value is still negative because it is still on the negative side of the y axis.
Inquiry Activity Reflection
1.) “The coolest thing I learned from this activity was" how SRTs can be used to find the 'Magic 3' ordered pairs of a Unit Circle.
2.) “This activity will help me in this unit because…” I now know how to derive the ordered pairs of the UC, so now I don't have to rely on my subpar memory when drawing it out.
3.) “Something I never realized before about special right triangles and the unit circle is…” the properties of the SRTs actually play a big part in deriving the ordered pairs of the UC.
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