I/D3: Unit Q Concept 1: Pythagorean Identities

1. Inquiry Activity Summary 

sin^2x + cos^2x= 1

Where does sin^2x+cos^2x=1 come from? 

• It comes from the Pythagorean theorem. The Pythagorean theorem is an identity, which is a proven factor or formula that is always true. (Kirch) The Pythagorean theorem is an identity for that same reason since it has been proven and it is always true. 

What is the Pythagorean theorem using x,y, and r? 

• We know that the Pythagorean Theorem is a^2+b^2=c^2.
• When we plot it in coordinate plane using the unit circle we get x^2+y^2=r^2. "x" being the horizontal leg, "y" being the vertical leg of the triangle and "r" being the hypotenuse which is equal to one since we are dealing with the unit circle.

Pythagorean theorem equal to 1

• In order to have the Pythagorean Theorem equal to 1 we divide by r^2 in order to give us the product of 1. The equation would now be x^2/r^2 + y^2/r^2 =1 which can be simplified to a nicer form of -
  

The ratio of cosine on the unit circle is x/r 

The ratio for sine on the unit circle is y/r. 

What do you notice? 

• We notice that what is being squared in our previous equation are the values that we have just found out. They are the values of what sin stands for and what cos stand for that are being squared and the value of r is one because that is the length of the hypotenuse always one on the unit circle. 

We can conclude:

• We can conclude that sin^2x + cos^2x= 1 is an identity as well because it has been proven and is always true because it follows the Pythagorean theorem method.

How do we know it is true? 

• We plug it in which demonstrates that the equation is true, it is valid since it is equal to 1 just like the equation had stated it would be. 
• making this an identity since it has been proven and it will always be true. 

Derive to identify with secant and Tangent

What do we know?

sin^2x + cos^2x =1

• we simplify this in a way that will give us both secant and tangent in only one step
• in order to get this we will divide everything by cos^2x 
• Which will end up giving us sin^2x/cos^2x + cos^2x/ cos^2x= 1/cos^2  
• which can be simplified to tan^2x + 1 = sec^2x 

Derive to Identify Cosecant and Cotangent 

We know sin^2x + cos^2x = 1 is where we are starting from 

•  we first look at what is being asked for us so we can solve in just one step
• in order to get both cosecant and cotangent into the equation we must divide by sin^2x
• this will end up giving us 1+ cot^2x= csc^2x which has all that is being asked to have 

2. Inquiry Activity Reflection 

1. The connections that i see between units N,O,P so far are that they are all coming from the unit circle, all triangles eventually end up connecting to the unit circle and a right triangle. Also how in units N,O, and P there are connection between knowing the trig ratios, because they will come in handy the further along we get into the unit. Knowing the trig functions help us better understand what is going on and where everything is coming from.

2. If i had to describe trigonometry in three words it would be complex, rigorous, and interesting. This is so complex because there are many parts to it and it test what we have learned previously. It is rigorous because not only is it not learned in a short amount of times but we have to understand the connection between everything in order to fully understand the concept. Interesting, because once a light bulb goes off in your head the world finally makes sense and  you feel like you can conquer the world.