SEMESTER PROJECT

The Pythagorean theorem can be used to derive the distance formula. Given a²+b²=c², and given an x,y coordinate plane, let a be equal to the distance between the x-coordinates \(x_2-x_1\) and b be equal to the distance between the y-coordinates \(y_2-y_1\). Then, the hypotenuse c is equal to the distance between these two points, say d. Using these facts, the original equation could be rewritten as \((x_2-x_1\))²+\((y_2-y_1\))²=d². Taking the square root of both sides to isolate d gives \( \sqrt{(x_2-x_1)^2+ (y_2-y_1)^2} \)=d , which is the distance formula.

The theorem is also used in trigonometry. Consider the identities \( \sin^2 θ =1-\cos^2 θ \), \( \cos^2 θ =1-\sin^2 θ \), \( \sec^2 θ =\tan^2 θ +1\), and so forth. These identities are found using the unit circle with a radius of 1. Remembering that 1²=1, rearranging these identities gives \( \sin^2 θ +\cos^2 θ =1\), \( \cos^2 θ +\sin^2 θ =1\) , \( \sec^2 θ =\tan^2 θ +1\), and so forth. These identities are equivalent to the definition of the Pythagorean theorem, since they display some first term "a²", some second term "b²", and some third term "c²". The use of the Pythagorean theorem in trigonometry allows us to rewrite trigonometric expressions in equivalent terms that are more useful in another context. If we would like to convert between sine and cosine values of an angle without knowing the measure of the angle, the Pythagorean theorem with these trigonometric identities also allows us to do so (Khan Academy).

The Pythagorean theorem has many uses outside of mathematics classes. The Pythagorean theorem is used in architecture, construction, woodworking, and more. It can allow you to calculate the slope of a roof, and it can help construct square buildings using Pythagorean triples. Babylonians used the concept of a²+b²= c² and Pythagorean triples in surveying, which is still practiced to this ay. Surveyors use right triangles to calculate the steepness of mountainous slopes since the terrain in those areas is usually uneven.

The theorem can also be used for two-dimensional navigation to find the shortest distance between points given two separate lengths. Zamboni (2018) gives examples of sea and air navigation. At sea, the legs of the triangle would be the distance north or south and east or west, while the hypotenuse would be the shortest distance to the desired point, which is what you should follow. In the air, planes use their height above the ground and the distance from the airport to calculate when the plane should begin to descend. Additionally, one might see a beaten path across grass where people have stepped off the pavement. Their goal is to arrive at their destination quicker, and they achieve this by subconsciously applying the concept of the Pythagorean theorem! Instead of following the sidewalk (the ''legs'' of the triangle), they chose to create the path of the ''hypotenuse'' through the grass.

One of the most frequently proven theorems in history, the Pythagorean theorem is used across many aspects of our lives. The theorem has a long and deep history that spans several cultures and civilizations. While Pythagorean concepts were used over 3000 years ago, Pythagoras and his monastery provided a written proof of the theorem, although the original author is unknown. Today, the theorem is recited in schools, and most students can recall the theorem from memory (just be sure to include the hypothesis).