In the measuring your world project, there were many interrelated points we had to go through. Starting us off was understanding the Pythagorean Theorem. The formula is a²+b²=c². C in the formula represents the longest side of a triangle known as the hypotenuse. By adding together the next two remaining sides of the triangle (both to the power of 2), we would solve for the length of the hypotenuse. There are many ways to prove the effectiveness of the Pythagorean theorem and we explored this through multitudinous worksheets and papers outlining how and when to properly utilize the formula. We found it most effective when applying it to right triangles. We sometimes had to make our own right triangles by dividing regular triangles down the middle.
We can use the pythagorean theorem and what we learned from it to derive a distance formula. To find this formula, we relatively re-label the variables which comes out to: √(x2 + x1)2 + (y2 + y1). If we place a triangle on a grid and instead of a²+b²=c² we replaced it with √(x2 + x1)2 + (y2 + y1), we can figure out the distance between the points farthest away from each other.
When applied to a cartesian coordinate plane, which is also known as a unit circle where each point is equidistant to the locus of the circle. With right triangles, we can verify the equivalent distance between the farthest of a triangle point and the center. This works because all circles are similar so the same methods would still relevantly apply. On our common circle, the radius is always 1. The unit circle (shown below) is a helpful tool to determine which distance applies to certain angles on a cartesian coordinate plane. With the circle, we are given a rule: “out = 90 degrees minus the in”. |
With the unit circle, since it applies to a grid like the one shown above, we can recognize that each point on the circle is the uppermost point of an imaginary 90 degree triangle. The top point on a 30 degree and 90 degree triangle would be (square root 3 over two), (one over two).
Because all circles are similar, we can go across the quadrant plane's lines of symmetry and locate the remaining points on the circle. We can do this by using reflections of our original quadrant to the top right. After we've figured out the reflection of the original points onto the remaining four areas of the circle, we then can apply their positive or negative status to finish labeling them correctly.
As previously mentioned, the rule “out = 90 degrees minus the in," pertains to the unit circle. With this, we can use the unit circle diagram to figure out that cosine theta = sin(90 degrees minus theta). This is also commonly known as a trigonometric identity. To find the sine and cosine, we look at the circle and can judge from there the angles which we can input sine and cosine to find the proper side lengths of the triangles.
We've talked about sine and cosine, but what about tangent? Tangent Is equal to the opposite (triangle side) divided by the adjacent. This is in order to find the length of the hypotenuse which is fairly simple.
The equation above is in relation to the SOH CAH TOA we learned in class. Tangent is TOA. Cosine is equal to adjacent divided by the hypotenuse. Sine equals the opposite over the hypotenuse. Each equation closely relates to each other because using one, you can find an element that will be useful for another, but one must initially start with using one out of the three SOH CAH TOA equations and sine, cosine, tangent functions.
The unit circle proves to be a useful tool in this respect. With it, we can find arccosine, arctangent, and arcsine. These are when there is a negative 1 involved in the problems we're solving. We are able to use the circle to find it and we actually used it on one of the SAT practice problems.
By using sine, cosine, and tangent, we can find side lengths to a triangle, as well as distance if it is measured as such. By using two points, we can use sine or cosine to find the lengths of the distance from one point to another. We used this in the Mount Everest problem by creating a triangle where one point was the peak of the mountain and we used sine and cosine to find the distance from our two points to the pinnacle of Mount Everest. This further clarified the concept of determining distance. We also learned how it could be applicable to daily life.
When learning the law of sines, I initially struggled with the concept and plugging it into genuine math equations, but over the span of the few weeks we've been working on it, I feel as if I have significantly enhanced my knowledge of the subjects.
Similarly, this applied to learning the law of cosines because you can input a number for sine and by putting a second number in for cosine, they will be equivalent. It was very helpful to learn all three functions at once, because now, I see how they relate to each other and their very strong ties and connection. If you are missing one, you could always go back and use the other two equation functions in SOH CAH TOA. Over all, this was a challenging and confusing, yet educational topic to learn.
Because all circles are similar, we can go across the quadrant plane's lines of symmetry and locate the remaining points on the circle. We can do this by using reflections of our original quadrant to the top right. After we've figured out the reflection of the original points onto the remaining four areas of the circle, we then can apply their positive or negative status to finish labeling them correctly.
As previously mentioned, the rule “out = 90 degrees minus the in," pertains to the unit circle. With this, we can use the unit circle diagram to figure out that cosine theta = sin(90 degrees minus theta). This is also commonly known as a trigonometric identity. To find the sine and cosine, we look at the circle and can judge from there the angles which we can input sine and cosine to find the proper side lengths of the triangles.
We've talked about sine and cosine, but what about tangent? Tangent Is equal to the opposite (triangle side) divided by the adjacent. This is in order to find the length of the hypotenuse which is fairly simple.
The equation above is in relation to the SOH CAH TOA we learned in class. Tangent is TOA. Cosine is equal to adjacent divided by the hypotenuse. Sine equals the opposite over the hypotenuse. Each equation closely relates to each other because using one, you can find an element that will be useful for another, but one must initially start with using one out of the three SOH CAH TOA equations and sine, cosine, tangent functions.
The unit circle proves to be a useful tool in this respect. With it, we can find arccosine, arctangent, and arcsine. These are when there is a negative 1 involved in the problems we're solving. We are able to use the circle to find it and we actually used it on one of the SAT practice problems.
By using sine, cosine, and tangent, we can find side lengths to a triangle, as well as distance if it is measured as such. By using two points, we can use sine or cosine to find the lengths of the distance from one point to another. We used this in the Mount Everest problem by creating a triangle where one point was the peak of the mountain and we used sine and cosine to find the distance from our two points to the pinnacle of Mount Everest. This further clarified the concept of determining distance. We also learned how it could be applicable to daily life.
When learning the law of sines, I initially struggled with the concept and plugging it into genuine math equations, but over the span of the few weeks we've been working on it, I feel as if I have significantly enhanced my knowledge of the subjects.
Similarly, this applied to learning the law of cosines because you can input a number for sine and by putting a second number in for cosine, they will be equivalent. It was very helpful to learn all three functions at once, because now, I see how they relate to each other and their very strong ties and connection. If you are missing one, you could always go back and use the other two equation functions in SOH CAH TOA. Over all, this was a challenging and confusing, yet educational topic to learn.
Part 2
Sydney and I decided to measure a common number 2 wooden Ticonderoga pencil. For this, we selected a random pencil from Sydney's pencil bag and began to measure it. We chose a pencil because it has three very basic geometric shapes which we could find the areas of and add up. On it, is a cone, a cylinder, and a hexagonal prism which we added together later. Our approach was very simple. Below is our mathematical process for finding the areas of each of the three shapes.
A challenge we faced was deciding what object to measure. Initially, we decided to measure the circumference of the Earth using cross referenced data from google and various sources. That proved to be a little too complicated or a little too simple depending on the parameters we set for completing it. We also had a challenge with measuring in decimals because we had to continuously double and triple check our work in order to make sure the measurements and data was as accurate as possible and would work for what we needed. A success we had was working together efficiently and inputing the equations. We both understood the questions and ways to go about solving them so with our thorough knowledge, we diligently completed all of our work while maintaining high quality standards. To measure the pencil, we used the habit of a mathematician of taking things apart and starting small and being systematic by separating the pencil into three primary shapes then finding the volume of each and after that, we added them together. Overall, this project was very successful for us. |