The first part of the measuring your world project was proving the Pythagorean theorem. The Pythagorean theorem is a ² + b² = c². We proved it by dividing a right triangle into two similar triangles. We then saw that if the sides are lined up the hypotenuse of the larger triangle would equal two sides of the smaller triangle. We then learned about the distance formula using the Pythagorean theorem. We formed a right triangle between two points and since a² + b² = c² you could use the difference in x and y to calculate the distance between two point or the hypotenuse. We used this distance formula to find the equation of a circle. We found this by putting the unit circle in the center of a coordinate place and using the distance formula to find its radius. The unit circle is a circle with a radius of one. We found three specific point on the unit circle, one at 30 degrees, one at 45 degrees, and one at 60 degrees. We found these point because these form special triangles. We could use the circles symmetry to find the point for the corresponding lines in the other four quarters of the circle. We could then use the unit circle to find sine and cosine. The sine function finds what the y would be on a unit circle and cosine solves for the x. The tangent function can be found using a radial. The tangent and radial are always at 90 degrees to each other. You can use similarity to see how sine cosine and tangent relate to each other. Using this you can learn that sinθ = Opposite / Hypotenuse, cosθ = Adjacent / Hypotenuse, tanθ = Oposite / Adjacent. We can then use the unit circle to show arc sine, arc cosine, arc tangent. These are the inverse of what they where before. We then used the mount Everest problem to discover the law of sine's. We where given the angles and a side of a triangle and had to find the other sides. We could do this by using the law of sine's that shows how the sine of an angle divided by the side opposite it is the same in a triangle for all of its angles. We finally learned that the law of cosines is c² = a² + b² - 2abcosθ.
Measurement Presentation
For my measurement presentation I measured the volume of a sword blade and reshaped it while keeping the same volume. I started by getting all of the length measurements of the blade and converting them from inches to centimeters. Then I found the area of the tip of the blade and the rest of it and added them together. When reshaping it I decided to a 1 centimeter long edge at the top and see how much the height would have to change to keep the same volume. I started finding the area of the tip by finding the area of the two triangles at the sides. I used trigonometry to find their area and then added that to the area of the 1 centimeter long middle. After finding the volume of the tip I found what the volume the rest of the blade would need to be and since only the height would change I found what that number needed to be. That was how I had found what the blade would look like it was reshaped with the same volume.