BQ#5: Unit T Concepts 1-3

Whenever you have 0 in your denominator it will be undefined. You can only have an asymptote if you have an undefined number. All the trig functions have a ratio:

Sine               Cosine                  Cosecant                   Secant                  Tangent                Cotangent

      y/r                    x/r                           r/y                           r/x                         y/x                         x/y

When we are talking about the Unit Circle "r" matter what it equal to 1. When we look at the ratios, sine and cosine are the only trig functions where r is the denominator. This means no matter what the numerator is, the function will never be undefined because the denominator will always be 1. All the other functions have the possibility of being undefined because x or y can equal to 1 or 0 and that all depends on what point on the graph you are on. If you are on 0 or 180 or 360 degrees, cosecant and cotangent will be 0 because the ordered pairs on that point are (1,0) and 0 is y. If you are on 90 or 270 degrees, secant and tangent will be 0 because the ordered pairs are (0,1) and x is 0. 

BQ#2: Unit T Intro

1. The Unit Circle is a trig graph unraveled. It follows the same patterens as it does on the Unit Circle. When you have sine on the Unit Circle, it is going to be + + - -, on the trig graph it is the same thing it is up up then goes down down and that ends that period of the graph and on the Unit Circle it marks then end of a whole circle.

The period of sine and consine is 2pi because then you look at the negatives and positive from the Unit Circle, it takes one whole circle for the pattern of positives and negatives to start again. When you go around the Unit Circle completely that is 2 pi, that is why the period is 2pi. The reason why tangent and cotangent is pi is because it takes half the circle for the pattern to start over.

Sine and cosine are the only ones that have an amplitude, they are also the only two graphs that do not have that do not have asymptotes. All the other graphs have asymptotes because they certrain points where the graph is undefined and does not touch those points. With sine and cosine they graph will touch any points as long as they are within the amplitude. With all the other trig functions they have asymptotes. Amplitude and asymptotes are restrictions for each graph.

Reflection #1: Unit Q-Verifying Trig Identities

1. To verify a trig function is to show that the function given on the left side matches up the function given on the right side by simplify, substituting, and multiplying. You never touch the right side. The point is to get both sides to equal each other.

2. What has really helped me solve these functions was to memorizes all the identities. At first I thought it was pointless but when you are in the process of verifying or simplifying a trig function its a hassle to constantly having look back and forth. MEMORIZE! MEMORIZE! MEMORIZE!

3. The first thing I always do is try to change everything to sine and cosine. It makes everything a lot easier when it is in sine and cosine. Then I look for GCF. From there I look to cancel out anything and or combine like terms. From there I see if i can substitute an identity again and I'm usually done by this point.

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.