SP#7: Unit Q Concept 1: Using fundamental identities to simplify and verify expressions

Please see my SP7, made in collaboration with Chelsea Amezcua, by visiting their blog here. Also be sure to check out the other awesome posts on their blog.

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.

WPP#13&14: Unit P Concept 6-7: Applications with Law of Sines and Cosines

This WPP#13-14 was made in collaboration with Chelsea Amezcua. Please visit the other posts on their blog by going here.
1.)Amarie Alize and Steven met today at a mutual friends' Super Bowl party. Before the game started the host decided to have a football game among all the friends. However, no one could find the football. Everyone was trying to look for the football(which was to the north of them) and at the same time Amarie Alize and Steven(who are 25 yards apart) found it. Steven saw the ball's path to be N28E from a north-south line through where he is standing and Amarie Alize saw the ball's path to be N39W from a north-south line through where she is standing. What is the distance between Amarie and the football?

 2.)Steven and Amarie were playing football and they both run to catch the ball. Steven runs 12 yds at a bearing on 56 degrees and Amarie runs 14 yds at a bearing of 315 degrees. How far apart were they before they started running? 

BQ#1: Unit P Concept 1-2: Law of Sines SSA and Area of an oblique triangle

2. SSA is ambiguous because referencing to our unit circle sine can be positive in two quadrants that are less than 180(a triangle's angles add up to 180), which is the first and second quadrant(of the reference angle). Also, it is ambiguous because for AAS and ASA we had the angles or we only had to find one by using the triangle sum theorem.

4.The area of an oblique formula is derived from the geometry are of a triangle formula, which is 1/b*h. If we draw a triangle an we do not know it's  height in order to find the area, then we can make that triangle into two triangles by drawing a line from the top (m<B) to the middle(between m<A and m<C or just in the middle of side b). We can label that h because that will be the height of the triangle.Next, we can take the sin of m<A and it will be h/c (SOH=OPPOSITE/HYPOTENUSE). Then, we can multiply c to both sides for it can cancel in the denominator and we are left with csin m<A=h and now we can plug in h in A=1/2bh and we get A=1/2b(asinC). However, this can be written in many ways depending on what you are given. For example, A=1/2bcsinA;A=1/2acsinB;A=1/2absinC.

www.ck12.org/book/CK-12-Trigonometry-Concepts/r2/section/5.4/

References:http://www.ck12.org/book/CK-12-Trigonometry-Concepts/r2/section/5.4/

WPP#12: Unit O Concept 10: Solving angles of elevation and depression word problems

1. It is Fourth of July and Amarie Alize is looking up at the fireworks from ground level. She estimates the angle of elevation from where she is now to the fireworks up in the sky to be 23. She was watching the fireworks that her neighbors were exploding illegally. She knows the distance from where she stands to her neighbor house(where the fireworks are coming from) is 100 feet. How far up are the fireworks exploding?(Round to the tenth place)

http://www.bing.com/images/search?q=fireworks&go=&qs=ds&form=QBIR#view=detail&id=90331943C86B19A045E9F3CB827184526A6E3CCD&selectedIndex=12

2. Amarie Alize took some ski lessons during the winter, but instead of going to the rookie slope she went to the advanced slopes. She is just about to ski down a steep mountain. She estimates the angle of depression from where she is now to the bottom of the mountain to be 30. She read a sign that said "500 ft above ground level." How long is the path that she will ski?

http://www.bing.com/images/search?q=steep+mountain&qs=IM&form=QBIR&pq=steep+mountain&sc=8-14&sp=1&sk=#view=detail&id=2973C31EE05CF2709444D44E1B40C67FFD5D2CEE&selectedIndex=1

 

 

 

 

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.