1.) 30 degrees
There were several part that were needed to be labeled and solved for. The hypotenuse is equaled to 1, and due to the rules of the 30,60,90 right triangle: the side adjacent to 30 (which is x on the pic) has a value of x radical 3; the side opposite (which is y) has a value of x; and the hypotenuse (r) as 2x. To find an actual number value, trig functions (sin and cos) must be used. The sin of 30 degrees solves for the opposite side (y) and is found with y/r; substituting the variables will result with x/2x and simplified will equal to 1/2. The cos of 30 degrees solves for the adjacent side (x) and is found with x/r; substituting the variables will result with x radical 3/2x and simplified will equal to radical 3/2. If the triangle were to be placed in a Unit Circle, with the origin being located at the labeled angle measure, three coordinate pairs will be needed and two are found with the values solved. The origin is (0,0), while only going across the x axis is (radical 3/2,0), and going directly above from the previous coordinate is (radical 3/2,1/2).
2.) 45 degrees
The instructions are the same but differ in the number values. Due to the 45,45,90 right triangle rules, x is the side adjacent to the given degree with value of x; the y is opposite to the angle but similar to the adjacent side as it, too, has the value of x; and the hypotenuse, r, is equaled to 1 and x radical 2. The trig functions of sin and cos are also used with sin 45 degrees = y/r and cos 45 = x/r. Solving for the sin and cos is similar with this SRT since the same values are used to substitute, which is x/x radical 2 and simplified equaling to radical 2/2. If the triangle were to be placed in the Unit Circle, with the origin being located at the labeled angle measure, there would be three coordinate pairs. The origin would be (0,0), shifting to the right would be at (radical 2/2,0), and moving directly upward will get to (radical 2/2,radical 2/2)
3.) 60 degrees
The instructions and requirements are as well the same from the previous triangles, and it will be closely identical to the 30 degree triangle. Same as before: x is adjacent but with the value of x; y would be opposite to the angle with the value of x radical 3; and the hypotenuse, r, is 1 and 2x. Sin and cos will be used to find the length of the sides. The sin of 60 degrees (y/r) would be x radical 3/2x and simplified will equal to radical 3/2. The cos of 60 degrees (x/r) would be x/2x and simplified will equal to 1/2. If the triangle were to be placed in a Unit Circle, with the origin being located at the labeled angle measure, three coordinate pairs will be (0,0) at the origin, (1/2,) going across the x axis, and (1/2, radical 3/2,) shifting directly up.
4. This activity helps derive the unit circle by figuring out the certain point of the circle with the special right triangles. Figuring out the lengths of these triangles, if placed on the circle, will every time give the same coordinate pairs that are points on the Unit Circle if done correctly.
5.)
The quadrant in which the original triangles from the beginning are draw in the first one. Depending on which quadrant the SRT is found, the value will change by its sign (positive or not). If the triangle is found on Quadrant II, then x from the coordinate pair will be negative. In Quadrant III, both x and y will be negative. Finally in Quadrant IV, only y will be negative. In the three pictures provided above, the triangles in a different location from Quadrant I will change in its sign due new location. In Quadrant I, all trig functions are positive, while in Quadrant II only sin and csc are positive and the rest negative. Meanwhile in Quadrant III, tan and cot are the only positive. Finally in the Quadrant IV, only cos and sec are positive.
INQUIRY ACTIVITY REFLECTION
1.) “THE COOLEST THING I LEARNED FROM THIS ACTIVITY WAS..." that special right triangles are actually beneficial and do a bit more than just being a perfect shape.
2.) “THIS ACTIVITY WILL HELP ME IN THIS UNIT BECAUSE..." I can refer back to it if I ever forget how to fill out the Unit Circle for the coordinate pairs section.
3.) “SOMETHING I NEVER REALIZED BEFORE ABOUT SPECIAL RIGHT TRIANGLES AND THE UNIT CIRCLE IS…” how they relate to one another. I would have not have guessed that the coordinates on the Unit Circle could also come from different shape.
