1. The 30* Triangle
The 30* triangle is one of the special right triangles and it requires simplification in order to be able to label the triangle correctly. The first step to the simplification of this triangle is to label it correctly according to the rules of special right triangles. The labeling should be that the hypotenuse is 2x, the leg opposite of the 30* is x, and the longer leg of the triangle is radical 3. To simplify that we must divide the length of each leg and the hypotenuse by 2x because it is the only way our hypotenuse will equal 1, which is our "r." The leg opposite of the 30* should end up being 1/2, our "y," and the longer leg should end up being radical 3/2, the value of our x. When we graph the triangle in the unit circle we draw the coordinate plane and this triangle should be in the first quadrant. The origin of the triangle is (0,0) , going across the horizontal line of the graph we should have (radical 3,2), and when we go upwards from there we get (radical 3/2, 1/2).
2. The 45* Triangle
The 45* triangle follows the same rules as the 30 60 90 triangle when it comes to the labeling of the triangle. To simplify the 45 45 90 triangle you must divide by x, except for the two other legs, those must be divided by the reciprocal of 1/x so the x can cancel out and leave you with (radical 2/2) for both legs of the triangle. When graphing it you must use the coordinate plane and it should land in quadrant one, once again, and the origin of the 45 45 90 triangle is (0,0), when you move over on the horizontal line of the graph you should end up with (radical 2/2, radical 2/2), and you should end up with the same when moving up on the graph.
3. The 60* Triangle
The same rules as the 30* triangle applies to this one and the only difference is the value change. In this triangle the horizontal value is x and the longer leg is radical 3/2. The labeling still remains the same, the hypotenuse is still 1, the angle that is opposite of the 60* is still y and the horizontal line is 1/2x. When plotting the graph the origin is (0,0), the horizontal points should be (1/2.0) and when you reach the top point, it is (1/2, radical 3/2).
4. How does this activity help derive the unit circle?
This activity helped me learn that the unit circle is just a bunch of triangles and that they all make up the points in the unit circle, depending on which quadrant they lay on. The quadrant they lay on can determine whether both the x and y are going to be positive, only in quadrant one, in quadrant two only the x is negative, in quadrant 3 both the x and y are negative, and in quadrant 4 the y value is negative.
5. How do the values change if you draw the triangles in quadrants II, III, and IV?
The values change depending on the quadrants. If the triangle is in quadrant II then only the x will be negative, if the triangle is in quadrant III then both the x and y will be negative, and if the triangle is in quadrant IV then only the y is going to be negative.
Inquiry Activity Reflection
The coolest thing I learned from this activity was that all the points on the unit circle are repeated in each quadrant but with different values, like negative and positive.
This activity will help me in this unit because it helped me to understand where everything in the unit circle comes from and how to differentiate everything that's on the unit circle. With all this information I will be able to fill out the unit circle to help me on the test.
Something I never realized before about special right triangles and the unit circle is that they are both related to each other. I didn't know that the points in the unit circle came from the special right triangles.
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