I/D Unit N Concept 7 Unit Circle How do SRT and UC Relate?

Inquiry Activity Summary

1. 30* Triangle - This is the smallest angle that is opposite to the smallest side. The 30* angle can be rewritten as radians and is π/6. To convert the * to radians, you use a conversion shown as 30 x π/18. The coordinates for this on a plane are (√3/2,1/2) in quadrant I when plotting the Unit Circle. The unit circle is essentially a circle with the radius of 1. The reference angle is how far an angle is from the x intercept. For 30* angles and its coterminals(angles with the same reference angle), the reference angle is 30*.  

2. 45* Triangle - This is one of the special right triangles (isosceles) that has two sides that are the same angle (45*) and a 90* angle. For converting to radians, 45* x π/180 that results with π/4 in quadrant 1. The sides are in a ratio of 1:1: √2 In a unit circle, the coordinates would be ( √2/2, √2/2) in quadrant I. As for the reference angle, it would be 45* Angles with the same reference angle are 135*, 225*, & 345* that are within 1 revolution. 

3. 60* Triangle - The 60* triangle is part of the 30* triangle to make up the special 30/60/90 right triangle. The 60* angle is the largest angle between the 30/45/60*. In radians, 60* x π/180 = π/3 in quadrant 1. The ordered pair for a 60* angle is (1/2,√3/2) which is the inverse of the 30 ordered pair in quadrant I. The reference angle for a 60* triangle is 60*. For angles that have a reference degree of 60*, the angles are: 60*, 120*, 240*, 300* These angles are within the 1 revolution of a unit circle. 

4. This activity helps me derive the Unit Circle because the first quadrant is comprised of these angles and knowing these angles helps me figure out the rest of the quadrants in the unit circle. There are certain patterns like how much π increases each time or how the reference angles work in a way to make sense and the (+-) for coordinates are supposed to look like. By knowing the special properties of the 3 degrees and its radians, I am able to completely fill out the rest of the unit circle in under 5 minutes without any problems except for the occasional (+-) for the coordinates. 

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5. All of the angles mentioned above lie in the first quadrant of the unit circle. These values change by degrees, radians, and ordered pairs when drawn in different quadrants because of the special properties of the circles. Notice the 150* in quadrant 2, even though the reference angle may be the same as the 30* in the first quadrant, the new angle and its radians are completely different from the original 30*. Its ordered pairs may look the same but the x value has become a negative because the angle is in the left side (negative side) of the graph but the y value stays the same because it is still in the positive side of the graph. In the third quadrant, there is a 225* with the reference angle of 45* that is similar to the 45* in the first quadrant. However, like the 30*, even though the reference is the same, the angle, radians, and ordered pairs are completely different from the 45* in the first quadrant. As for the 300* in the fourth quadrant, the reference angle is 60* like the first quadrant but the 300*, the radian, and the ordered pair specified in the fourth quadrant are completely different from the 60* in the first quadrant. This is also seen when the angles with the same reference angle are drawn in all the quadrants and connecting them would make a quadrilateral in the unit circle. The values for each point on the unit circle are identical except for the (+-) which is determined by where the point is on the graph. With this, it helps determine the rest of the unit circle and its values just by using the first quadrant and the 3 special right triangles. 


Inquiry Activity Reflection

The coolest thing I learned from this activity was a single angle can have an infinite possibility of coterminal angles. 

This activity will help me in this unit because all the concepts revolve around knowing the unit circle and the special properties of the angles in the unit circle. The concepts that are most important in relation to the unit circle are 7,8,9. 

Something I never realized about special right triangles and the unit circle is how the special right triangle shows up in the unit circle and plays a significant role in filling out the rest of the unit circle and how the unit circle actually has a method for completing it when I all did last year was try to memorize everything. 

RWA 1 Unit M Concept 4 - Conic Sections in Real Life

1. Parabola - "Set of all points that are equal distance from focus and directrix."
2. Algebraically (x-h)^2=4p(y-k) & (y-k)^2=4p(x-h)
    Graphically -

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             The formula used for parabola is unique from the other conic sections because it only has term that is squared while the rest are not. As for the graph, it is unique in having one arc with points of all equal distance between the foci and directrix.

             Writing standard form - Decided whether the graph is x^2 or y^2, move the squared term and its like terms to one side and the others to the rest, complete the square correctly and make sure that the side that does not have the squared term is not negative. If it is take, take out the negative. 

             Graphing parabola - Standard form, plot vertex, plot the focus and directrix "p" units away from vertex, sketch axis of symmetry, sketch parabola. 

             "Eccentricity is how much a conic section deviates from being circular." For parabolas, the eccentricity should always equal to 1 or else you did something wrong. To determine the eccentricity of a parabola, you use the formula e=c/a. Your c & a are found by the standard form of the parabola. The foci is the distance from the center to the interior that gives the parabola its shape. The directrix is the line at the exterior of the parabola that is also the equal distance from the center. The foci and directrix determine if the graph is wide or thin because the points are all equal distance from the foci and directrix.

               Always make sure your formula is in standard form as well when graphing parabolas because it greatly helps for determining the required parts for the graph. Depending if P is positive and if the equation is x or y, will tell the person how the parabola will be graph. Ex. P is positive and the equation is x^2 means the parabola will open up. 

             After graphing parabola go through the checklist so that you can get full credit on your problem and not make any mistakes. Also make sure that the foci is inside the parabola while the directrix is outside of the parabola. Labeling each part is important so that you do not get messed up somehow.

3.          A real world use of parabolas are the parabolic skis. The skis make it much easier to turn and the turning area to help make a perfect arc with less skill. This skis are the improved versions of the original circular cut skis. Compared to the original skis, the original tend to only move in a straight line making it much harder and requiring more energy to make sharp turns. 

            These skis are more commonly used now because they provide more efficiency and require less energy. By forming perfect arcs and increasing the speed of the turns, the skiers are able to use little energy and save time on their runs for better times. Also, the parabolic skis offer more control which is essential for controlling the skis to be more efficient.  Parabolic Skis

            Here is a video to help people understand how the shape of the skis affect the riders and their times in their races and marathons. 

4. Citations
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