I/D#3: Unit Q - Pythagorean Identities Concept 1: Using fundamental identities to simplify and verify expression (simple, one or two step identities)

1. When you draw a triangle inside a Unit Circle we use Pythagorean Theorem to finding the missing sides. We can use the Pythagorean Theorem to show why it is an identity.

It is referred to a Pythagorean Identity because you are using the Pythagorean Theorem to solve for the triangle and we have our ratios at the end of the equations which show that it is sine and cosine.  

Now if you were to plug in any of the coordinate points from the 30 45 60 degrees reference angles, you will always get answer that is equal to 1. For example:

2. To get the second identity from the Pythagorean Identities you have to take what you got from the first identity which was sin^2x+cos^2x=1 and divide it all by cos^2x. From there you use your ratio and reciprocal identities to find what each divided problem is equal to. And from there you have your tangent and secant identity.

To get the third identity you once again take your first Pythagorean Identity which was sin^2x+cos^2x=1 and divide it all by sin^2x. You then have to multiple divided identities and you use your ratio and reciprocal identities to find what each one equals too and from there you have your final identity that uses cotangent and cosecant

SP#7: Unit Q Concept 2: Find all trig functions when given one trig function and quadrant (Using identities)

This SP#7 was made in collaboration with Stephanie Vargas.  Please visit the other awesome posts on their blog by going here.

    You can solve this specific problem several different ways and this is one of them. In this SP we show you how to solve using identities and also using SOHCAHTOA. Make sure that you read all the side notes so you are able to understand what is going on and why we do each step. Solving using identities is trickier so make sure you look over each step as it is being done and refer back to you identities to understand why each step is happening.

BQ#1: Unit P Concept 2 & 3: Law of sines SSA (one, none, two solutions). Law of Cosines SSS or SAS

2. Law of Sines

The triangle always has to be able to fit in a scale of 180 degrees. With the given sides and angle you have to determine whether it will be two, one, or none answer(s). When you are finding the prime sides/angles you will have to use the very first angle that you have found in order to do so. You use the reference angle from the first answer to find the prime answers, the prime answer will be in quadrant II. If all your angles add up to more than 180 on your prime side, you will have no second answer. If BOTH your answers are more than 180, you will have NO answers. If both answers add up to 180 exactly, you will have TWO possible answers.

4. Area formulas

         This will work with any side you have. The height has created a right triangle and this let's you use trig-functions. The height will be whatever you get for the sine of C times the side of a. 

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

WPP#13-14 was made in collaboration with Stephanie Vargas. Please visit the other fantastic posts on their blog by going here.

I/D# 1: Unit N Concept 7 : Knowing All Degrees and Radians Around the Unit Circle, Knowing All the Ordered Pairs Around the Unit Circle, Understanding and Applying ASTC to the Unit Circle

30 Degrees Triangle 

45 Degrees Triangle 

60 Degrees Triangle & The Unit Circle

The coolest thing I learned from this activity was… the special right triangles that we learned about in geometry (which was way back in freshmen year) is actually related to the Unit Circles we began learning about sophomore year. I now know that the ordered pairs aren't just numbers put together randomly, I understand where these numbers come from.
This activity will help me in this unit because…I understand where these numbers come from when it comes down to labeling all ordered pairs. I am able to figure out how to get these ordered pairs if I ever forget what goes where. I can just draw my triangles and solve them to find my ordered pairs.
Something I never realized before about special right triangles and the unit circle is…that they are related. When I first started learning about special right triangles I thought it was going to be something that we just learned and weren't going to go back on it and use it later on like we do when we learned algebra. We are actually taking what we learned about special right triangles in geometry and finding the ordered pairs of them for the Unit Circles we are now learning about in Math Analysis (Pre-Calculus).

RWA #5: Unit M Concept 5: Graphing ellipse given equations (must complete the square first) and identifying all parts (center, focus, major axis, minor axis, vertices, co-vertices, eccentricity)

1. Ellipse- The set of all points such that the sum of the distance of the two points is a constant


2. Equation


Graph
https://www.youtube.com/watch?v=lvAYFUIEpFI

3. RWA- "'The Whisper Chamber' at the United States capital is another example of an ellipse's special properties. Stand at one focus and you can hear perfectly someone whispering at the other focus, though it is much too far away to hear if not for the reflective property specific to ellipses." The National Statuary Hall is were the US House of Representatives use to meet. You are able to whisper from the focus point of the room and the other person on the other focus point is able to here. 
4. References

• http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet
• http://www.math.utah.edu/online/1010/conics/
• https://www.youtube.com/watch?v=lvAYFUIEpFI
• https://sites.google.com/site/arinsellipse/real-life-examples

WPP#9 Unit L Concepts 4-8: Calculating Possibilities

Create your own Playlist on MentorMob!

Pay close attention to the wording in each problem. the wording tells you what kind of problem its going to be and what formula to use