Measuring Your World
Part 1:
In the Measuring Your World unit we learned about several different formulas crucial to helping us solve geometrical problems, these were the eight trigonometric formulas. One of these formulas was sine. Sine is a trigonometric function that is equal the ratio of the side opposite a given angle, in a right triangle, to the hypotenuse. The formula for sine is: sin(angle) = opposite/hypotenuse.
The second formula that we learned was the formula for cosine. The formula for this function is: cos(angle) = adjacent/hypotenuse.
Another formula that we were taught in this unit was the formula for tangent or tan. This function has a little bit of a different formula from the last two: tan(angle) = opposite/adjacent.
The ArcSine function is basically the inverse of the sine function. This means that instead of using an angle to solve side lengths, you use side lengths to solve for the angle. You use sine when finding side lengths and ArcSine for when you want to find the angle. This is the same with the ArcCosine and ArcTangent functions as well. The way that ArcSine is written is: sin^-1.
The second inverse function is ArcCosine. ArcCosine is the inverse of cosine and as I stated for arcsine, practically does the opposite or vice versa of what cosine does. This is written as cos^-1.
ArcTangent is the inverse function of tangent. Written as: tan^-1.
The Law of Sines is sin(A)/a = sin(B)/b = sin(C)/c. Law of Sines is an equation that relates the lengths to the sines of the triangle’s angles. This is only when given two angles and one side length.
The Law of Cosines is when given two side lengths and one angle. As far as I know the formula for the Law of Cosines is: (a-b * cos(angle))2 + (bsin(angle))2 (The twos being exponents.)
The Pythagorean Theorem (a^2 + b^2 = c^2) is a very important formula because it helps to prove whether or not a triangle is a right triangle, acute, or obtuse triangle. If the equation works and the hypotenuse squared is equal to the side lengths squared then it is a right triangle. If the hypotenuse squared is longer than the side lengths squared then it is an obtuse triangle. And finally, if the hypotenuse squared is shorter than the side lengths squared then the triangle is acute.
The Distance Formula is similar to the Pythagorean Theorem and can be used to find the measurement from one point to another. This is very important to solving for distance between two points and can be used in a variety of different situations. This formula has proven very useful because it is also used to help solve for the Pythagorean theorem as well.
In using the distance formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane we found that you can make a circle by placing points the same distance in different directions away from the center, the more points you add the more it could be perceived as a circle. We incorporated the distance formula because you can use that to determine the distance from one point to another, helping us solve for the circle. This was proven to be very useful on several other problems when trying to find the distance between two points on a shape other than a triangle.
The unit circle is a circle with a radius of one that we used to help us define trigonometric functions. This was very useful for practice and was used a lot in several of our class assignments.
Trigonometric functions can be used when finding points on the unit circle by creating a triangle from the center to the chosen point, then straight downwards. You can find a different point depending on what number of degrees you use. After choosing a point you can find the point by using sin, cos, or tan. Also by using x2 + y2 = 1 because the circle is a unit.
Since a circle has one radius and doesn't have any longer or shorter parts of it, meaning it is equal on both sides, if you solve for a point on one side that should be the exact same measurement for the corresponding point on the other side because they are mirrored or equal sides. However, some numbers on other sides may be negative if using a graph.
Using the unit circle you can make a right triangle which we can use to define, using the angle of theta, sine and cosine. By using the angle of theta we can determine that sin(angle theta) = o/h and that cos(angle theta) = a/h. By using x2 + y2 = 1 (x being cos(angle theta) and y being sin(angle theta)) we can define sine and cosine.
Another function that we learned about in this unit was the tangent function. The tangent function when used in right triangles is when the tangent of an angle is the length of the opposite side divided by the length of the adjacent side, (tan(angle) = O/A).
In the Measuring Your World unit we learned about several different formulas crucial to helping us solve geometrical problems, these were the eight trigonometric formulas. One of these formulas was sine. Sine is a trigonometric function that is equal the ratio of the side opposite a given angle, in a right triangle, to the hypotenuse. The formula for sine is: sin(angle) = opposite/hypotenuse.
The second formula that we learned was the formula for cosine. The formula for this function is: cos(angle) = adjacent/hypotenuse.
Another formula that we were taught in this unit was the formula for tangent or tan. This function has a little bit of a different formula from the last two: tan(angle) = opposite/adjacent.
The ArcSine function is basically the inverse of the sine function. This means that instead of using an angle to solve side lengths, you use side lengths to solve for the angle. You use sine when finding side lengths and ArcSine for when you want to find the angle. This is the same with the ArcCosine and ArcTangent functions as well. The way that ArcSine is written is: sin^-1.
The second inverse function is ArcCosine. ArcCosine is the inverse of cosine and as I stated for arcsine, practically does the opposite or vice versa of what cosine does. This is written as cos^-1.
ArcTangent is the inverse function of tangent. Written as: tan^-1.
The Law of Sines is sin(A)/a = sin(B)/b = sin(C)/c. Law of Sines is an equation that relates the lengths to the sines of the triangle’s angles. This is only when given two angles and one side length.
The Law of Cosines is when given two side lengths and one angle. As far as I know the formula for the Law of Cosines is: (a-b * cos(angle))2 + (bsin(angle))2 (The twos being exponents.)
The Pythagorean Theorem (a^2 + b^2 = c^2) is a very important formula because it helps to prove whether or not a triangle is a right triangle, acute, or obtuse triangle. If the equation works and the hypotenuse squared is equal to the side lengths squared then it is a right triangle. If the hypotenuse squared is longer than the side lengths squared then it is an obtuse triangle. And finally, if the hypotenuse squared is shorter than the side lengths squared then the triangle is acute.
The Distance Formula is similar to the Pythagorean Theorem and can be used to find the measurement from one point to another. This is very important to solving for distance between two points and can be used in a variety of different situations. This formula has proven very useful because it is also used to help solve for the Pythagorean theorem as well.
In using the distance formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane we found that you can make a circle by placing points the same distance in different directions away from the center, the more points you add the more it could be perceived as a circle. We incorporated the distance formula because you can use that to determine the distance from one point to another, helping us solve for the circle. This was proven to be very useful on several other problems when trying to find the distance between two points on a shape other than a triangle.
The unit circle is a circle with a radius of one that we used to help us define trigonometric functions. This was very useful for practice and was used a lot in several of our class assignments.
Trigonometric functions can be used when finding points on the unit circle by creating a triangle from the center to the chosen point, then straight downwards. You can find a different point depending on what number of degrees you use. After choosing a point you can find the point by using sin, cos, or tan. Also by using x2 + y2 = 1 because the circle is a unit.
Since a circle has one radius and doesn't have any longer or shorter parts of it, meaning it is equal on both sides, if you solve for a point on one side that should be the exact same measurement for the corresponding point on the other side because they are mirrored or equal sides. However, some numbers on other sides may be negative if using a graph.
Using the unit circle you can make a right triangle which we can use to define, using the angle of theta, sine and cosine. By using the angle of theta we can determine that sin(angle theta) = o/h and that cos(angle theta) = a/h. By using x2 + y2 = 1 (x being cos(angle theta) and y being sin(angle theta)) we can define sine and cosine.
Another function that we learned about in this unit was the tangent function. The tangent function when used in right triangles is when the tangent of an angle is the length of the opposite side divided by the length of the adjacent side, (tan(angle) = O/A).
Part 2:
The object that we chose to measure for this project was a pig. We thought that it would be interesting to simplify and measure the volume of a live animal even though obtaining the measurements might have been slightly difficult. We decided that in order to effectively find the volume, we would have to divide the object into several different parts then add the separate volumes together to find the total. The way that we divided it up was the four legs, head, snout, and body.
The different formulas that we included were: tan72 (trigonometric formula), (area of an elipse) Aelipse = π ab, (area of a circle) Acircle = πr^2 (area formulas), and (volume of a sphere) Vsphere = 4/3 π r, (volume of a cylinder/oval prism) V = Abase x h (volume formulas).
Finding the volume of the legs consisted of using the formula for cylinder volume: Vcylinder = πr^2 * h, which eventually (using the measurements), lead us to find that the volume of the legs were: Vlegs = 94.24777961 in^3, which is the answer after we multiplied by four.
For the volume of the head we used the equation: Vsphere = 4/3 πr^3. The answer that we came to was: Vsphere = 179.59438 in^3.
Since the snout is not a perfect sphere, we used this equation: Aelipse = πab. We found that the volume was: Vsnout = 58.90486225 in^3 (this is the volume after being multiplied by its height, 5.)
We used this equation for the body: Aelipse = πab. Eventually coming to the conclusion that the volume of the body was Vbody = 4099.77841293 in^3, after being multiplied by its height of 30 inches.
The total volume of the pig that we found in the end was Final Volume: 4,432.525... in^3.
I think that in this project a majority of the group members we on task the whole time we were given to work on it. I really appreciate how we came together to work on some of the math section especially because anytime someone had a problem, the others would offer their help and overall I think that everyone was very willing to help complete the assignment. We did of course run into some small challenges like getting certain people to work and getting the measurements of the actual pig was no piece of cake, however we did eventually find a way to work everything out in the end.
The habits of mathematician that were used throughout this assignment were starting small and taking apart and putting back together. With starting small we really focused on every piece of the pig that we had divided. We put in a good amount of work into each piece to figure out it’s exact volume. Taking apart and putting back together factored in when we actually divided the pig into several different simplified pieces to help us to better understand its volume. We did this to simplify what we were doing and to focus on one piece at a time instead of making things more complicated than they needed to be.
The only thing that we would have done differently would be to possibly factor in other pieces of the pig to make the problem more complex or maybe not simplify some of the pieces as much as we did, it would have been a little more challenging but we would’ve learned a lot about time management. Another thing that we may have changed would have been dividing the work more evenly because some stuck with the things they were good at rather than branching out of their comfort zone more often.
All in all I would say that this assignment was a success and that I did learn a lot from it.