We first were introduced to this project with the Pythagorean Theorem c^2 = a^2 + b^2, showing how it relates to coordinate planes with the distance formula, (d=√(x2- x1)^2+(y2- y1)^2) to the hypotenuse of a triangle (c^2). We also use a Unit Circle on coordinate planes, the radius being one from the edge of the circle to the coordinate plane origin, to figure out its side lengths (x and y) for angles 45, 30, and 60 using the Pythagorean Theorem or reflection/rotation with a formed special right triangle.
Pythagorean Theorem is used while finding missing side lengths of a right triangle when you already have 2 of the lengths. This works because the sum of the squares created from the triangles square lengths match the formula. The a and b square’s areas add up to c’s area. If you split up the squares into separate triangles they are all equal in size. The Distance Formula is almost considered a version of Pythagorean Theorem because if both are drawn out on a graph they go through similar processes.
Pythagorean Theorem is used while finding missing side lengths of a right triangle when you already have 2 of the lengths. This works because the sum of the squares created from the triangles square lengths match the formula. The a and b square’s areas add up to c’s area. If you split up the squares into separate triangles they are all equal in size. The Distance Formula is almost considered a version of Pythagorean Theorem because if both are drawn out on a graph they go through similar processes.
While Pythagoras works with any right triangle and two given lengths, distance works for two points on a graph when given two points.
The equation of a circle is used to find a point when given the radius. This equation is similar to Pythagorean because you use similar a similar setup. However here you are solving for one point on either the y or x axis. Using the radius as the hypotenuse, you solve for the remaining side lengths and use those to find your point on the circle. We used the distance formula to prove that this is accurate because it is an identical concept of right triangles on a graph. You do all of this with a Unit Circle. A Unit Circle is a circle with a radius of one.
The equation of a circle is used to find a point when given the radius. This equation is similar to Pythagorean because you use similar a similar setup. However here you are solving for one point on either the y or x axis. Using the radius as the hypotenuse, you solve for the remaining side lengths and use those to find your point on the circle. We used the distance formula to prove that this is accurate because it is an identical concept of right triangles on a graph. You do all of this with a Unit Circle. A Unit Circle is a circle with a radius of one.
Using the Unit Circle, you can derive Sine and Cosine. In the Equation of a Circle, you solve for the x and y coordinate, and are given a specific angle. Cos is the x coordinate and Sin is the y, proven by finding the sin and cos of the angle then plugging them into Pythagorean theorem. You get 1, the radius of the Unit Circle. Then you can also discover than the Tangent is Sine/Cosine. Then you can find their functions from similar triangles. If you have two right triangles, both with the angle theta then all three of their angles must be the same, making them similar. One angle is 90, another is theta and the third is 90-theta. If you compare side lengths between the two triangles and consider theta as your main angle you get opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent. If you use a right triangle and do these equations it ends up being sin=o/h (soh), cos=a/h (cah) and tan=o/a(toa).You also discover that cos(theta)=sin(90-theta). The arc functions (arcCos, arcSin, arcTan) are used to find an angle from it’s lengths. This is basically a reversal of the normal functions, using o/h instead of sin(theta).
The final portion of this part of the project is the Law of Sines and the Law of Cosines. The Mount Everest Problem is about one question: How did surveyors find the peak of Mount Everest using 2 theodolites? The theodolites and the peak create a triangle with only one given side length and two known angles. Both of these equations are derived from a triangle that is not right. The Law of Cosines is used when you have two side lengths and one known angle while Law of Sines is used when you have two angles and one side length. Law of Cosines is closer to Pythagorean Theorem and the Distance formula whereas Sines is similar to the tan, cos and sin functions. Both end up with all angles and side lengths solved, so the only difference is what you are given at the start. We learned about the Law of Sine by working with triangles that were not right triangles, meaning that the pythagorean theorem could not be applied. One problem we solved using this method was the Mount Everest problem, in which we split into to right triangles by dropping a perpendicular and then solving for the missing length. We were able to use this method because in order to use the Law of Sines two, or three, of the angles given and one of the side length to solve for the missing sides. Written out the law of sines is sinB/b = sinC/c = sinA/a. The law of cosines is used when we know two lengths and one angle between them, this is written as c2 = a2 + b2 – 2ab cos C. Each of these formulas can be applied to any problem with or without a right triangle that requires looking for a length or height.
The final portion of this part of the project is the Law of Sines and the Law of Cosines. The Mount Everest Problem is about one question: How did surveyors find the peak of Mount Everest using 2 theodolites? The theodolites and the peak create a triangle with only one given side length and two known angles. Both of these equations are derived from a triangle that is not right. The Law of Cosines is used when you have two side lengths and one known angle while Law of Sines is used when you have two angles and one side length. Law of Cosines is closer to Pythagorean Theorem and the Distance formula whereas Sines is similar to the tan, cos and sin functions. Both end up with all angles and side lengths solved, so the only difference is what you are given at the start. We learned about the Law of Sine by working with triangles that were not right triangles, meaning that the pythagorean theorem could not be applied. One problem we solved using this method was the Mount Everest problem, in which we split into to right triangles by dropping a perpendicular and then solving for the missing length. We were able to use this method because in order to use the Law of Sines two, or three, of the angles given and one of the side length to solve for the missing sides. Written out the law of sines is sinB/b = sinC/c = sinA/a. The law of cosines is used when we know two lengths and one angle between them, this is written as c2 = a2 + b2 – 2ab cos C. Each of these formulas can be applied to any problem with or without a right triangle that requires looking for a length or height.