I/D3: Unit Q - Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:

The formula in this unit is sin^2x+cos^2x=1. Thinking back to the unit circle, the Pythagorean Theorem is x^2+y^2=r^2. However, when put into the triangle in the unit circle, x=cos and y=sin and for "r" to be 1 we have to divide everything by r^2. We are then left with (x/r)^2+(y/r)^2=1 (are those ratios looking familiar?). Th ratio for cosine is (x/r) and the ratio for sine is (y/r), so we substitute cosine and sine for x and y respectively. That is how the formula sin^2x+cosine^2x=1 is derived. 

a) Deriving the identity with Secant and Tangent. Deriving from the sin^2x+cos^2x=1. Because we are looking for secant and tangent, which is the reciprocal of cosine and the ratio for tangent is y/x (sine over cosine), I would divide everything by cosine so that 1/cosine would be secant and the sine over cosine would be tangent.

b) Deriving the identity with Cosecant and Cotangent. Deriving from the sin^2x+cos^2x=1. Because we are looking for cosecant and cotangent, I would divide everything by tangent because sin because x/y (cosine over sin) is the ratio for cotangent and 1/sin is cosecant.

INQUIRY ACTIVITY REFLECTION:

• The connections that I see between Units N, O, P, and Q so far are the Pythagorean theorem used from the Unit Circle from the Unit N and O can help us derive the formulas for Unit Q. Anther connection that is used throughout the units is the ratios, no matter what side length it is or if its a triangle not from the unit circle, the ratios remain the same throughout.
• If I had to describe trigonometry in THREE words, they would be intimidating, confusing, and rewarding. Intimidating and confusing sort of combine together because once you first start trigonometry, you feel overwhelmed by the information so you feel confused. However, I also describe it as rewarding because then you start learning various ways to remember the formulas and find out how derivation can help you if you forget certain formulas. It's rewarding in a sense that you learn different ways to learn formulas and a way to use your knowledge to derive the formula out of information you already know.