I/D #1: Unit N Concept 7: How Do SRT And UC Relate?

Inquiry Activity Summary

[Kelsea Del Campo; worksheet]

1. The 30* triangle is one of the special right triangles that is taught to us in Geometry. This means that each side has a special equation that goes with it. The hypotenuse is 2x, the horizontal angle is x, and the vertical angle is x radical 3. Now, we have to identify which is the x and y. The horizontal is x and the vertical is y. The hypotenuse is r, which refers to being a reference angle, the angle from the terminal side to the closest x axis. Now if we set the reference angle equal to one, we have to do that to all of the other sides by dividing by 2x. This means that we have to divide by 2x on the horizontal and vertical side. This means our hypotenuse will equal 1, our vertical angle equal to 1/2, and our horizontal angle equal to radical 3 over 2. Next I drew a coordinate plane where the triangle was in the first quadrant. Then I labeled the vertices into ordered pairs. The edge is (0,0) since it is at the middle of the graph. The one to its right is ( radical3/2, 0) because the x is radical3/2.  The one above it is (radical3/2, 1/2). The last ordered pair is a point on the unit circle and its angle is 30*. This can help us identify the ordered pair on the unit circle with a reference angle of 30*, depending on which quadrant one or both will be negative. 

[Kelsea Del Campo: worksheet]

2.  The 45* angle is also a special right triangle. The hypotenuse is x radical2, the vertical side is x, and the vertical side is also x. Now we also have to identify x and y. X will be the vertical side and y will be the horizontal side. The hypotenuse is, again, r. We have to set the hypotenuse equal to 1 again so we divide all three sides by x radical2. After doing so, our r should equal 1, our y should equal radical2/2, and our x should equal radical2/2. Next draw a coordinate plane with the triangle in the first quadrant. Label the verices as ordered pairs. The edge is (0,0), the one beside it is (radical2/2, 0), and the one above it is (radical2/2, radical2/2). The last ordered pair is a point on the unit circle with an angle of 45* and can help us find the ordered pairs for an angle with a reference angle of 45*, depending on which quadrant one or both will be negative. 

[Kelsea Del Campo: worksheet]

3. The 60* angle is very similar to the 30* except that everything is switched. This means that the vertical side is x radical3 but remains y. The horizontal is now x but remains the x. The hypotenuse remains the same. Set the hypotenuse equal to 1 by dividing by 2x and divide the rest by 2x as well. Now the hypotenuse should equal 1, the y should equal radical3/2, and the x should equal 1/2.Next, draw a coordinate plane where the triangle is on the first quadrant. Then identify the vertices as ordered pairs. The edge should be (0,0), the one beside it should be (1/2,0) and the one above it should be (1/2, radical3/2). As before, the last ordered pair is a point on the unit circle and is the same ordered pair for that with a reference angle of 60*, depending on which quadrant one or both will be negative.

4. This activity helps us derive from the unit circle because when we look at the unit circle and look at all reference angles of 30*, it has the same ordered pair, depending on the quadrant a negative will be present. Any angle with a reference angle with 45* has the same ordered pair, depending on the angle there will be a negative on the x or y, or both. Lastly, for any angle with a reference angle of 60*, it will have the same ordered pair, and depending on the quadrant, the x or y or both will be negative.

[Kelsea Del Campo: phone]

5. The triangle drawn in this activity all lie in the first quadrant. If we draw the triangle in the second quadrant, it will be a mirrored image of the first quadrant, the third quadrant will be a mirrored image of the second quadrant, and the fourth quadrant will be a mirrored image of the third quadrant. The closest angle to the x-axis is going to be an angle with a reference angle of 30*, the middle will that with a reference angle of 45*, and the one closest to the y-axis will e the one with the reference angle of 60*. The x's in the second quadrant will be negative, x and y will be negative in the third quadrant, and the y in the fourth quadrant will be negative. The 45* angle is shown in the second quadrant and is like a mirror image of the one in the first quadrant. It has the same ordered pair but its x is negative. The 30* angle is in the third quadrant and is upside down and the x and y of the ordered pair is negative. Lastly, the 60* angle us also upside down but only the y is negative.

[Kelsea Del Campo: phone]

[Kelsea Del Campo: phone]

Inquiry Activity Reflection

1. "The coolest thing I learned from this activity..." was how the special right triangle corresponds are are very similar in each quadrant. I didn't learn in Algebra II where these numbers came from so that was also interesting.

2. "This activity will help me in this unit because..." we are currently memorizing the unit circle to help us the sin, cos, and tan so it will help in finding the answer. 

3. "Something I never realized before about special right triangles and the unit circle is ..." that we get the ordered pairs from the special unit circle when we set the hypotenuse equal to 1 and that the triangle can be found in the unit circle itself.