I/D#3: Unit Q Cocept 1: Using fundamental identities to simplify and verify expressions(simple,one or two step identities)

INQUIRY ACTIVITY SUMMARY

1.)

     a. An identity is a proven formula and fact that are always true. The Pythagorean Theorem is an identity because it is a proven formula that is true.

     b. The Pythagorean theorem  is x^2+y^2=r^2.

     c. If you wanted x^2+y^2=r^2 to equal 1 you will need to divide r^2 to the other side and you are left with x^2/r^2 +y^2/r^2=1. However, we can rewrite that and say (x/r)^2+(y/r)^2=1 because of the distributive property of a power.

     d. The ratio for cosine on the unit circle is x/r.

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

     f. We can plug in cosine for x/r in the Pythagorean theorem(x/r)^2+(y/r)^2=1 and sin for y/r. We can conclude that the Pythagorean theorem is cos^2theta+sin^2theta=1.

    g. Sin^2theta+cos^2theta=1 is referred to as a Pythagorean identity because we can plug in cosine for x/r in the Pythagorean theorem (x/r)^2+(y/r)^2=1 and sin for y/r and it is still the Pythagorean theorem but in words because the trig function for cos was x/r and sin's  trig function was y/r. Therefore, it is the same as (x/r)^2+(y/r)^2=1 because we know the trig function for cos was x/r and sin's  trig function was y/r.

     h.

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2.)

     a.
     b.

INQUIRY ACTIVITY REFLECTION 

1. The connections that I see between Units N, O, P, and Q so far are the unit circle provided us with the ordered pairs of trig function and then we wan use those trig functions to solve for missing angles and sides of special right triangles and non-right triangles and now we can use the unit circle trig functions and ordered pairs to rewrite the Pythagorean theorem. Perhaps, we can use those trig functions and the Pythagorean theorem to now find missing angles and sides of special right triangles.

2.If I had to describe trigonometry in THREE words, they would be algebra geometry combined.

I/D#2: Unit O Concept7-8: Deriving the patterns for the 45-45-90 and 30-60-90 special right triangles

INQUIRY ACTIVITY SUMMARY

#1-3 are performed in the video

We cut the first triangle diagonal because it will form a triangle  no like if you cut it straight it will give you a rectangle. We cut the second triangle straight because it will form a triangle and not another shape when cut horizontal or diagonal.

INQUIRY ACTIVITY REFLECTION 

1. Something I never noticed before about special right triangles is that we can get the patterns for special right triangles not by memorization, but by the Pythagorean theorem.

2.Being able to derive these patterns myself aids in my learning because I do not have to cram in another pattern into my brain for it can remember it and if I freeze during the test and can't remember which pattern goes with what triangle I can actually take a couple of seconds and find the patterns by using the simple Pythagorean theorem and not have to fail my test. 

I/D#1: Unit N Concept 7-9: Knowing all degrees,radians,and ordered pairs around the unit circle;understanding and applying ASTC to the unit circle;finding exact values of all 6 trig functions when given angle in degrees or radians;finding angles when given exact value of any 6 trig functions

INQUIRY ACTIVITY SUMMARY

Numbers #1-3 are preformed in the video.

4.This activity helps you in deriving the unit circle because the "y's" vertice(the top vertice) ends up being the ordered pair for  the 30,45, 60 degree in the unit circle and those are you main repeated ordered pair of the unit circle, except the other ordered pairs are mirrored. 

5. The triangle drawn in this activity are located in quadrant 1. For quadrant 2, quadrant 1 is mirrored to the left, but all of the x values are negative. For quadrant 3, quadrant 2 is mirrored down, but both the x and y values are negative. For quadrant 4, quadrant 1 is mirrored down, but the y values are negative.

In quadrant 2, the 30 degree triangle just like folded or mirrored to the left, which made the x value become negative because looking at it as in a coordinate plane, the left is in the negative. In quadrant 3 the 45 degree triangle mirrored to the left, but then mirrored down. Therefore, both the x and y values are negative because it is down and left. In quadrant 3, the 60 degree just mirrored down or if you want you can see it as if it mirrored to the left, then mirrored down, and finally mirrored to the right to make a complete circle. The x value remained positive, but the y is in the bottom right, which is negative.

INQUIRY ACTIVITY REFLECTION

1. The coolest thing I learned from this activity was the unit circle is formed by 30,45,and 60 right triangles and the unit circle's ordered pairs are all the vertices of those triangles.

2.This activity will help me in this unit  because first I will not be mandated to remember all of the ordered pairs and if  I decided to just remember the ordered pairs and I forget the ordered pairs of the unit I could just see the triangles and find their vertices and those will be my ordered pairs.

 3.Something I never realized before about special right triangles and the unit circle is that the triangles vertices make up the unit circle and before I never applied special right triangles to anything I would just do the soh cah toa with special right triangles and now the special right triangles are applied to the unit circle.