For this unit in math, we used many formulas, equations, and proofs to find the dimensions of 2-d shapes. We also used may different shapes such as right triangles, equilateral triangles, isosceles triangles, circles, squares, and rectangles. We started off with the basics, which led us into something new every time, which ultimately led us to the end which tied everything together.
First, we proved the pythagorean theorem while working with right triangles. The pythagorean theorem is a squared + b squared = c squared. Essentially, the two smallest sides of the right triangle squared and added together, equaled the longest side of the right triangle squared. We used the pythagorean theorem to then derive the distance formula. The distance formula is AB=√(x2−x1)2+(y2−y1)2. This formula allows one to find the distance between two different points on a cartesian coordinate plane. By finding the distance between two points, you are finding the length to the hypotenuse of a right triangle. This then allows you to use the pythagorean theorem to find the other side lengths of the triangle. Now, you can use the distance formula to derive the equation of a circle centered at the origin of a cartesian coordinate plane. The equation is x squared + y squared = r squared, with r being the radius. The two points (x and y) are squared and if they equal the radius squared, that point is a unit on the circle.
Next, we learned about the unit circle. The unit circle is a circle that is centered a the origin (0,0) of a cartesian coordinate plane. It has an infinite amount of points while also having a radius of one. There are also certain points that correspond with certain angles, allowing one to use those numbers to solve side lengths. We used the unit circle to create right triangles and solve side lengths/ angles. We also had to find certain points on the unit circle that corresponded with their respective angles. We use the equation, x squared + y squared = 1 or -1. The numbers 1 and -1 are used as the unit circle only reaches that far. Next, we used the symmetry of the unit circle to find the remaining points on the circle. For example, the 45 degree point is (√2/2, √2/2), which means the opposite point that has an angle of 225 degrees is (-√2/-2, -√2/-2). We also learned about how to define sine and cosine through the unit circle. Sine is opposite/hypotenuse and cosine is adjacent/hypotenuse so by using the radius of the unit circle, you can find the side lengths of the triangle which give you the point on the unit circle. You also used the angle theta while solving the trigonometry, which essentially was just the angle that corresponded with the point you were finding.
The next topic we learned was tangent. We first did a proof which showed that tangent is a perpendicular line that touches the circle at a point but does not cross that point into the circle. Then we learned about the tangent function and how it relates to triangles. The tangent function is opposite/adjacent. Next, we learned about deriving the sine, cosine, and tangent, through similarity and proportions. We learned this by using trigonometry law practice which showed us that by plugging in angles and side lengths into the general functions, you could solve unknown lengths without even using a triangle. We also worked with triangles that weren't right triangles, such as isosceles and equilateral triangles. In this case, we made our own right triangles and solved one right triangle, which solved the rest of the proportional right triangles. We also learned about the functions arcSine, arcCosine, and arcTangent which are the inverse functions of sine, cosine, and tangent. These functions are used to find the angles of theta 1 and theta 2 on a right triangle, which added with the right angle, equals 180 degrees. Depending on which side lengths you are using, you use a specific function. All three inverse functions use the same side length sequence as their respective counterparts. This can also be used to find the angle which corresponds to a certain point on the unit circle.
At this point, we went into the law of sines and worked with it through the Mount Everest problem. We took apart the triangle that surveyors mapped and used the law of sines to figure out the missing side lengths. We separated the triangle into two right triangles, figured out the angles of those two triangles, and found the height of the triangle using h=c*sinB after learning that the law of sines is sin(A)/a = sin(B)/b = sin(C)/c. This is also how we learned about the law of sines since they connected to one another. Then we learned about the law of cosines. These are the three laws in the law of cosines: 1. a squared = b squared + c squared - 2*b*c*CosA. 2. b squared = a squared + c squared - 2*a*c*CosB, 3. c squared = a squared + b squared - 2*a*b*CosC.