I/D3: Unit Q - Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:  

1)What is an identity? Why is the Pythagorean Theorem an identity.? 

An identity is a proven fact or formula which is always true. The Pythagorean theorem is an identity in that whatever numbers you plug in you're able to get the correct numbers.

b) What is the Pythagorean Theorem using x y and r?

Since our original theorem used a^2+b^2=c^2 With triangles being inside our unit circle we can replace a with x, b with y and c with r which results in x^2+y^2=r^2

c)

Here we divide by r to set our equation equal to one

d) What is the ratio for cosine on the unit circle?

x/r

e) What is the ratio for sine on the unit circle?

y/r

f, g and h)

What do you notice about part c in relation to parts D and e? What can you conclude?

Looking at our very top equation in this picture we take our value of the ratio for cosine and sine of the unit circle and plug it in and what we get is that our equation consists of is taken from the unit circle. Its also derived from our new transformed formula. Our sinx^2+cos^2x=1 is referred to as an identity because whatever number you plug in will result in one respectively. Using 45 45 degrees I plugged the numbers into our equation and got 1. 

2) Perform a single operation fairly to derive the identity with secant and tangent. 

Here we take our equation and divide it by cos in which results the first part results in one and our second part results in tan (because we memorized it) and our 1 over cos is a reciprocal which results in it equal to sec.

b ) Perform a single operation fairly to derive the identity with Cosecant and Cotangent.

Here we divide by the value of sin in which results it in being one for our first part. Our second part it results in being cot, and our third part which is a reciprocal leads it to being csc.

INQUIRY ACTIVITY REFLECTION:

1)The connections I see between units N, O, P, and Q are how we use triangles within the unit circle, and the use and value of trig functions such as y/r= sin and etc and how the values can be taken the root and lead into the unit circle.

2) If I had to describe trigonometry in three words, they would be : headaches are present.