Scribe Post DC and Bee- June 4th

Hi People!

Well, today we started our Pre-Calculus class with the following POD:

θ is 4th quadrant and cos θ= 4/5. Find the values of csc θ and tan θ.

In other words, given that an angle has cosine= 4/5, we should, with that information and trigonometry skills we learned earlier in the year, solve for the cosecant and tangent of such angle.

Remember that:

cos θ= x/r
sin θ= y/r
tan θ= y/x
sec θ= 1/(cos θ)= 1/(x/r)= r/x
csc θ= 1/(sin θ)= 1/(y/r) = r/y
cot θ= 1/(tan θ) = 1/(y/x)= x/y















We know that,
cos θ= 4/5, so if cos θ= x/r, then for this problem x=4 and r=5. (Refer to bubbleshare graph)
And,
To find the values of csc θ and tan θ, we notice that we need to find the value of y.
Now,
To solve for y, we use the Pythagorean Theorem which is: x^(2)+y^(2)=h^(2)
For this graph, we can see that the hypotenuse= radius =5, and we already have x=4. So we can plug in and solve for y.

4^(2)+y^(2)=5^(2)
16+ y^(2)=25
y^(2)=9
y=√9
y=± 3

In this case,
y= -3, since our angle is in the 4th quadrant and y is always negative in the third and fourth quadrants. (Refer to bubbleshare)

So,
We can now say that since csc θ= r/y then, by plugging in, csc θ= 5/-3.
And, since tan θ= y/x, then tan θ= -3/4.


SUMMATION NOTATION


During today’s class, we learned a new topic called “Summation Notation”, aka “Sigma Notation”.


To introduce the topic to the class, Mr. A. gave us the following example problem:
∞
∑ 3(0.8)^(k-1)
k=1

Know that,
Sigma is the Greek letter ∑ which means “add the following” or “the sum of”
Summand is the expression being added. (In this problem the summand would be 3(0.8)k-1.)
Index is the variable that is changing. (In this case the index=k, but it can be expressed as any other variable. Ex: n, x, i etc.)
Limits of Summation are the upper and lower bounds on the index. They are specified on the top and bottom of ∑. (In this problem, the limits of summation are (1, ∞))

Therefore,
The problem is asking us to find the sum of the series 3(0.8)^(k-1)

To understand what this problem means, there is a more literal way to explain it.
It is an Expanded Form of the Summation of the series.
For the problem above, it goes as follows:

3(0.8)^(1-1) + 3(0.8)^(2-1) + 3(0.8)^(3-1) +3(0.8)^(4-1) +…

This means that,
Summation Notation is a simplified expression which shows us the sum of a given series.

______________________________________________________________________

The class was then given the following problem:

For the expression:

6
∑ -2k
k=1
We were asked to:
a. Write in expanded form
b. Evaluate the expression
c. Identify the summand.

To solve the problem,
We use the explanation previously given as a guide.

To write the expanded form of the expression given,
We have to take the summand and replace the index, which is k, with the limits of summation, which are (1, 6). This will look like:

-2(1) + -2(2) + -2(3) + -2(4) + -2(5) + -2(6)

Notice that,
We do not write three dots (…) at the en of the series because it is finite.

To evaluate the expression, or find the sum,
We use previous knowledge of sequences and series to evaluate it as the sum of a series (Sn).

Since,
-2(1) + -2(2) + -2(3) + -2(4) + -2(5) + -2(6)

Our series equals,
= (-2) + (-4) + (-6) + (-8) + (-10) + (-12)

We can prove it is an arithmetic series by subtracting tn-1 from tn and at time we solve for d.

-4 – (-2) = -2
-6 – (-4) = -2
-12 – (-10) = -2

So,
We have t1= -2 and d= -2

In order to find the S6 , we need to find t6 .

To do so,
We use the arithmetic sequence formula ( tn=t1+ (n-1)d )
T6= -2 + (6-1)(-2)
T6= -1

Now,
We have everything we need to find S6 with the formula Sn=(n/2)(t1+tn) .

S6= (6/2)(-2+ -12)
S6=3(-14)
S6= -42

Finally,
To find the summand we identify in the problem the expression being added.
In this case it is -2k.


The following problem is another Summation Notation problem but a little bit more complex than the one before.

It is the following:

For tn=3(0.8)^(n-1), find the sum of the 5th through 18th terms.

Written in Summation Notation form would be:
18
∑ 3(0.8)^(k-1)
k=5

Just by looking at the problem, we can identify the geometric series formula, which is t1(r^(n-1)), expressed as 3(0.8)^(k-1).

Therefore,
The ratio of the sequence is 0.8.

We use that information to find t5, which we will use as the first term of the series because, since we are asked to find the sum of the 5th through the 18th term, we don’t need terms 1-4.

We use the following formula to find t5:

tn= t1(r^(n-1))
t5=3(0.8^(5-1))
t5=3(0.8^ (4) )
t5=1.2288

The expanded form for this series is:
3(0.8)^(5-1) + 3(0.8)^(6-1) + 3(0.8)^(7-1) +3(0.8)^(8-1)+…+3(0.8)^(18-1)

Since we are going to treat t5 as t1, we have to use t18 as t14 for the sum to be reasonable.

By having t5 as t1,
We can now plug it into the formula to find the sum of any geometric finite series which is:

Sn= t1 ((1-(r^n)) / (1-r))
S5-18= 1.2288((1-(0.8^14)) / (1-0.8))
S5-18≈ 5.87


SIGMA METHOD
In order to advance our knowledge in Summation Notation and learned new ways to answer complex problems, we first have to learn several tools (formulas) of Sigma Notation.

The first rule we learned in class today is really simple. It goes as follows:

1. Remove the constant of multiplication ( c ) where ai is some expression dependent on i (i = index).

Some, an example of this method is:

n n
∑ cai = c ∑ ai
i = 1 i = 1

In this example,
c equals the constant of multiplication and ai is the expression depending on i. You notice how c has been factored out of the expression of summation, leaving only ai within the expression. According to this, we find the sum the n terms and then multiply it by the constant.

An example for this method can be:

6 6
∑ -2k = -2 ∑ k
k=1 k=1

*This means that we add up 1+2+3+4+5+6 and then, multiply the sum by -2.

You can see that by PRACTICING and evaluating, the Sigma Notation is a simplified and much simpler way of expressing the sum of a given series.

HOMEWORK: Read Pgs. 506-507
Problems on Pg. 508 # 1-15 ODD
* Remember to study hard for the Pre-Calculus Quiz this Friday (recomended not to miss class that day) and study for the FINAL EXAM!!!
Next Scribe: MAC and Nabil