Math Mistakes Add.& Sub. Complex Numbers; Link: http://mathmistakes.org/?p=1070

What went wrong?
The mistake that occurred in the first box is that they did not distribute the negative properly. They were also subtracting numbers but then kept the numbers they had subtracted. Another mistake that they did was that in the second box the student foiled the equation when it should have been subtracted.

What they needed to do?
They needed to distribute the negative sign properly and then combine like terms, as shown on the picture below. This problem should not have involved multiplying it involved adding and subtracting only. 

Math Mistakes Sum and Difference;Link: http://mathmistakes.org/?cat=374

   

What went wrong?
The mistake for all the three boxes are that the student did not use the sum and difference formula to solve the equations. They simply look at the Y or X on the degree ordered pair and add or subtract them to get there answer. This is incorrect...

...What they should have done:
What they needed to do was solve the equations with the sum  and difference formulas. This means each degree being used  is labeled as the U or the V and the ordered pairs are separated into  sin,cos,and tan to make it simpler to imput into the equations, as seen in the pictures below. The highlighted problems are the ones that are shown on the Math Mistakes( each colors corresponds to each equation).

Answer to Question

These responses reveal that the student doesn't understanding trig because he/she is just using the X and Y parts of the degree ordered pair instead of the sum and difference formula. He/She needs to study the formulas and separate each ordered pair in order to successfully solve these problems.

Math Mistakes Special Right Triangles; Link: http://mathmistakes.org/?p=1003

What went wrong?
The mistake was that he rationalized the 1 instead of equaling it to N√2 and solving it out that way. He used the Pythagorean identity  which is a big NO NO when solving this equation.

Correction: What he had to do was equal the 1 to N√2 because that is the corresponding formula for that side. As seen below each side has a corresponding formula, therefore when the correct way was to equal the 1 to N√2 which gives you 1/√2. Next you rationalize and you get the answer of  √2/2.

Comment: 

Using the Pythagorean identity is the wrong formula to use, this type of problem does not use this type of formula. This is not used because each side has a formula. If I were in a situation if I forgot the formulas for the sides, I would probably would have used the Pythagorean identity as well.

Math Mistakes: Combining Like Terms; Link: http://mathmistakes.org/?p=953

What does the picture show?

This picture shows an equation of combining like terms. The equation seen above has been solved wrong  but the words in red go through the correct process.

What the viewer needs to pay close attention to in order to understand the concept?

The viewer needs to pay close attention to what the mistake of the student was. The mistake he made was combining all the terms together even though they weren't alike. They combined the numbers and placed the exponent along with the X. 

Correction: What the student had to do was combine  like terms of the problem which were the numbers with the exponent of x^4(10x^4 & 8x^4). Then they just bring down the 12x^5 to what we had combined and the correct answer would be 12x^5+18x^4.

Comment:

When I looked at the picture I could instantly see the mistake. This problem could be easily solved, but he made a cheeses bucket mistake!!

Unit T Big Question #2

How do the graphs of sine and cosine relate to each of the others? Emphasize the asymptotes in your response 
a. tangent
b. cotangent
c. secant
d. cosecant

What is this concept about?

This concept explains how sine and cosine are related to making the graph of the other trig functions. Ratio and reciprocal identities are used in order to understand this concept.

What the viewer needs to pay close attention to in order to understand the concept?

We need to pay close attention to the pattern of the signs in the graph. There is a trick that involves the signs to know whether the drawings on the graphs are correct for cotangent and tangent. For sin and cos the graph will lie in the corresponding ratios place. For example sin will stay with csc and cos will stay with sec.

Sine and Cosine are used to draw the other trig ratio graphs because they use the ratio identities and reciprocal identities to know the asymptotes to draw there graph. 

Tangent/Cotangent

The signs we are using are of sine and cosine of each quadrant:

Quadrant 1- Positive ÷ positive =positive

Tan,Cot,Sin,Cos fall in positive.

Quadrant 2- Negative ÷ positive=negative

Sin positive; Cos,Tan,Cot negative 

Quadrant 3- Negative ÷ negative= positive

Sin and Cos negative; Tan and Cot positive

Quadrant 4- Negative ÷positive=negative

Cos positive; Sin negative, Tan and Cot negative

The trick was used above. What it is is that the division we do from sin and cos gives us the sign (positive or negative) in which the  tangent and cotangent lie in.

The picture above shows sin.cos,tan.cot being graphed. You can see the trick visually on the graph. 

CSC/SEC

The reciprocal will lie in the same position to its corresponding trig ratio.

Quadrant 1-

Sin,Cos,Csc,Sec fall in positive

Quadrant 2-

 Sin positive Csc positive; Cos, Sec negative

Quadrant 3-

Sin,Cos,Csc,Sec negative

Quadrant 4-

Cos,Sec positive,Sin, Csc negative

Unit T Big Question#3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill? Use unit circle ratios to explain.

What is the concept about?

This concept is going to explain how tangent goes uphill and cotangent goes downhill. It will clarify why these two graphs have the different motion yet having the same pattern.

What the viewer needs to pay attention to in order to understand concept?

The viewer needs to pay attention to where the asymptotes are drawn for cotangent and tangent. This determines the way the graph will be shaped. They may have the same pattern, but the graphs are shifted differently when drawing them. We also need to remember the positive and negative pattern because we are going to use it.

Explanation:

There is a difference between the shapes of the graph because of  where the asymptotes are placed for tangent and cotangent. 

Tangent:

The tangent has the ratio identities of sin/cos. Since the asymptote is placed at pi/2 and 3pi/2 the graph is drawn going upward. The pattern is positive and negative, positive and negative. The negative is going upward to the next period which is positive, then the next period starts negative going up to positive. This gives it the uphill motion.

Cotangent:

The cotangent has the ratio identities of cos/sin. This means the asymptote are located in the 0 and pi. The pattern is positive and negative, positive and negative. This period starts at the positive and then goes down to the negative period. Then again it starts at the positive and goes down to the negative period. This gives it the downhill motion. 

Unit T Big Question #4

Why does sin and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain?

Asymptote: A line that approaches the given curve but makes no contact with it. 

What is this concept about?

This concept explains why the cosine and sin do not have asymptotes and the other trig ratios do. We will be using ratio reciprocals and ratio identities from Unit Q to explain why trig ratios have asymptotes.

What the viewer needs to pay close attention to in order to understand the concept?

The viewer needs to pay attention to how each distinct place  is undefined on the graph, in other words where the asymptotes are placed when graphing. They also need to pay close attention to whether it is being drawn positive or negative in there period. Pay attention the explanation below on why sine and cosine don't have asymptotes.

Sine/Cosine

Sine does not have an asymptote because the ratio is y/r, R always equals one on the unit circle, therefore it can never be undefined. Same goes with Cosine it's ratio is x/r, the R is always one, so its not undefined. What both of these do have is an amplitude of one which was explained in Big Question #1 part B.

The pictures below explain how ratio identities and reciprocal identities are involved. It as well shows drawings of the asymptotes. 

Tangent

Tangent has a ratio identities of sin/cos, so whenever cosine equals zero the tangent is undefined leading it to have an asymptote. The asymptote on the graph is located wherever cosine equals 0 on the unit circle, in this case pi/2(90deg.) and 3pi/2(270deg.). 

   

Cotangent

Cotangent has the ratio identities of cos/sin, so wherever sine equals 0 the cotangent is undefined leading it to have an asymptote. The asymptote is located where the sin equals 0 , in this case at 0(0deg.) and pi(180deg.). 

Secant

Secant is the reciprocal of cosine, this is known from the reciprocal identities sec=1/cos. So similar to tangent whenever cosine equals 0 it is undefined leading it to an asymptote. The asymptote on the graph is located wherever cosine equals 0 on the unit circle, in this case pi/2(90deg.) and 3pi/2(270deg.). 

Cosecant

Cosecant is the reciprocal of sine and we know this by the reciprocal identities of csc=1/sin. Similar to cotangent whenever sin equals zero is where the asymptote is located.  The asymptote is located where the sin equals 0 , in this case at 0(0deg.) and pi(180deg.).