I/D #3 Unit Q Pythagorean Derivations

1. Where does where sin^2x+cos^2x=1 come from to begin with (think Unit Circle!). You should be referring to Unit Circle ratios and the Pythagorean Theorem in your explanation.

Sin^2x +cos^2x=1 comes from the Pythagorean Theorem. Since Sine is equal to y over r and that is eqaul to y^2. Cosine is equal to x over r, which in referring to the Pythagorean Theorem is eqaul to x^2. When you put this together you have x/r^2 + y/r^2 = 1 or sin^2x + cos^2x = 1.

When deriving the two remaining Pythagorean Theorems you have Tan^2theta +1 = Sec^2theta and 1 + Cot^2theta = Cosecant^2theta. For the first one you have tanx= sinx/cosx then you multiply tanx by tanx=sinx/cosx times sinx/cosx you will be left with Tan^2x = Sin^2x/Cos^2x. For the second one you will do the same but have One/tan and one/sin to work with instead of the other two.

2. The connections that I see between Units N, O, P, and Q so far are…

The connections that I see so far are that the unit circle expands so much farther than I thought. I see that the ratios of Sin,Cos, and Tan can be used in many numerous equations across the unit of trigonometry.

3. If I had to describe trigonometry in THREE words, they would be…

Difficult and Tedious

WPP # 13 & 14 Unit P Concept 6 and 7

This WPP post was made in collaboration with Joshua Nolasco. Please visit the other awesome posts found on their blog and our WPP by clicking here.

WPP # 12: Unit O Concept 10 Elevation and Depression Word Problem

A) Bob at ground level measures the angle of elevation to the top of the mountain's slope to be 70 degrees, if at this point Bob is 18 feet away from the building, what is the height of the mountain?

B) Bob now stands atop. He measures the angle depression from where he is now to the ski resort to be 48 degree. He knows that he is 700 feet higher than the base of the course. How long is the path that he will descend?

I/D #2 How can we derive the patterns from our special right triangles?

Inquiry Activity Summary

In class I completed a worksheet in which I had to find as to why and how I would find the special patterns of the two special right triangles. One of them being the thirty, sixty, ninety, and the other being the forty five forty five ninety triangle.

For the forty five, forty five, ninety triangle I drew a dotted line diagonally down the middle of the square. A square has a ninety degree angle cutting this down the middle creates a forty five degree angle. After you have done this you create the special triangle. Label your special triangle correctly. Label the shared side as Y, then label X . You are given a side length of one to work with. After you label both of your sides with the given, you use the Pythagorean Theorem to find the length of the hypotenuse. You should set this up as one squared plus one squared equals c squared, as an answer you should get radical two as an answer.  From this we derive the special pattern N, N, N radical 2. The reason for n is due to the fact that it can stand for any given value in the world.

For the thirty, sixty, ninety triangle I drew a dotted line straight down the middle of the triangle to split it in half. I label the shared line with Y and then I label X. You are given a side length of one to label the sides with.  When you do this you end up with the angles of thirty, sixty, and ninety. When this happens the side length that you were given of one changes to one half. You now have one and one half to work with for the Pythagorean Theorem You set it up with one squared plus b squared equals one half squared. You should end up with the answer of radical three over two for the side of Y. Now we can add N into the pattern. When we do this you can derive the special pattern you now have 2N, N, and N radical three. The reason for N is because it can stand for any number in the world, it has infinite values for it.

Inquiry Activity Reflection

Something I never noticed before about special right triangles is the N can stand for any value in the world.

Being able to derive these patterns myself aids in my learning because I can now fully understand how and where these patterns come from.