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.

I/D2: Unit O- How to Derive Special Right Triangles

Inquiry Activity 
1. Deriving a 45-45-90 triangle
 Click Here

2. Deriving a 30-60-90 triangle
Click Here

3. The reason why both of these the 30-6-90 triangle and the 45-45-90 triangle have variables of "n" and not just the numbers because there is a relationship between as to why these numbers are there and they end up being a patter for all triangles of that kind. The variables make it a rule and make it true for all the special right triangles.

Inquiry Activity Reflection

1.Something I never noticed about special right triangles is... that all the sides have a pattern that is common between that type of right triangle. For example, In a 45-45-90 triangle we know that "a" and "b" will be the same since they have the same angle value. The hypotenuse however has an extra radical 2 being multiplied to the value of both "a" and "b". For a 30-60-90 triangle we see that the length of the shortest side which could be classified as "a" has a value of n. The value of the medium size side is n radical 3, and the side of the hypotenuse is 2n. Which makes since we see that the hypotenuse is suppose to be greater since it is the longest side and it is in the equation.

2. Being able to derive these patterns myself aid in my learning because... it helps me understand how everything connects and allows me to fully understand how to solve each and every problem. This can be very important to me in the long run because if i ever forget the rules i can easily derive them using the Pythagorean theorem and will help me connect everything else to the unit circle and not just with a radius (hypotenuse) of 1 but of any number and can solve it.

I/D #1: Unit N Concept 7-9 : Applications of a right triangle apply to the creation of the unit circle

Inquiry Activity Summary
1. Video on how to find a 30 60 90 triangle.

2. Video on 45 45 90 

The values of r( hypotenuse ) is 1, the value of x(horizontal) is radical 2 divided by 2 , and y(vertical) is also radical 2 divided by 2.

3. Video on 60 30 90 triangle

4. This activity helps derive the unit circle by being able to just focus on one quadrant and then realizing that there are only certain things that change. This also helps see what the unit circle consists of, where we get our points from (the coordinates) and how everything eventually ends up changing just by the sign. This activity helped me understand what each coordinate meant on the unit circle by just focusing on one part and then managing to move what I have learned and understand why and how it changes.

5.How does everything tie together?

The sides for the 30 degree triangle will still be r=1 x=radical 3/2 and y= 1/2. For the 45 degree triangle r=1 y=radical 2 divided by 2 and y= radical 2 divided by 2. The 60 degree triangle will be r=1 y=1/2 and x= radical 3 over 2.

Inquiry Activity Reflection 

1. The coolest thing i learned from this activity is how to connect everything that we have learned in previous years to what is essential to learn now.

2. This activity will help me in this unit because it is the foundation as to where the unit circle comes from and the rational behind it. Which means that if i were to ever forget what the unit circle is i would be able to solve it and not just remember nonsense. This is also essential because these are the most important points of the unit circle that repeat throughout.

3.Something that i never realized before about special right triangles and the unit circle is that they actually have a relation between them. The entire unit circle consist of numerous right triangles that give it its coordinates. Without the special right triangle this would not be accurate or as easy to manage as it is now. The special right triangles are the fundamental aspects of the Unit Circle.