I/D3: Unit Q - Pythagorean Identities

Inquiry Activity Summary:
1. Where does sin^2x+cos^2x=1 come from?
This identity is derived from the Pythagorean theorem, and that is why it is known as a Pythagorean identity. Drawing from our knowledge about the unit circle, the Pythagorean theorem using x, y, and r would be: x^2+y^2=r^2. From here, we would perform an operation that makes the Pythagorean Theorem equal to 1, which is part of the identity. This would then result in: (x/r)^2+(y/r)^2=1. A pattern that we notice is that the unit circle trig ratio for cosine is x/r, while the unit circle trig ratio for sine is y/r. As a result, we can plug these two trig functions into our equation, and thus we derive the Pythagorean identity.
2. Show and explain how to derive the two remaining Pythagorean identities from sin^2x+cos^2x=1.
To derive the two remaining Pythagorean identities, we can perform similar operations:

• For the first derivation, we divide the equation by cos^2x. This will leave us with several functions that can be converted using trig identities. Cos^2x/cos^2x cancels out as 1, sin^2x/cos^2x is equivalent to tan^2x (as seen in a ratio identity), and 1/cos^2x is equivalent to sec^2x (as seen in a reciprocal identity). This leaves us with 1+tan^2x=sec^2x.

• For the second derivation, we divide the equation this time by sin^2x. This will leave us with several functions that can be converted using trig identities. Cos^2x/sin^2 is equivalent to cot^2x (as seen in a ratio identity), Sin^2x/sin^2x cancels out as 1, and 1/sin^2x is equivalent to csc^2x (as seen in a reciprocal identity). This leaves us with cot^2x+1=csc^2x.

Inquiry Activity Reflection:
The connections that I see between Units N, O, P, and Q so far are that they all deal with triangles and angles in some way. We are learning how to solve different kinds of triangles, from right triangles to acute and obtuse ones that we needed to apply the Law of Sines and Cosines to. We are using the trigonometric functions derived from triangles (sin, cos, tan, etc.) and using them as identities or to find angles in the unit circle. Another connection that can be seen between these four units is that most of them deal with the unit circle--for example, we just derived the Pythagorean identities using trig functions from the unit circle and can prove them using points from it.
If I had to describe trigonometry in three words, they would be complicated, challenging, and rewarding to learn. It is complicated because there are many derivations that we must learn and, more importantly, understand if we want to grasp a concept. Everything that we learn is built on something else, and it is important to understand these things before we move on to something new. Also, trigonometry is challenging because there are many formulas, many identities to memorize in order to succeed. Memorization is the biggest and most challenging part, because once we know the material well, we will be able to do well on the tests. Finally, trigonometry is almost rewarding to learn because I enjoy the feeling of success that comes after solving a triangle or simplifying a trig function using identities. It may be difficult at first, but once I know the concepts well, I am very satisfied with learning them.