The insult to the industry on the right is where my art career started. Although far from elegant, this little guy does do one thing quite well; he expresses simplicity. Displayed here is the equation r=(a)sin(b)theta with some minor tweaks. This most glorious display of craftsmanship was created only after the most intense mathematical training. Few are able to achieve the skill set required to construct such beauty. A complicated combination of r=(a)sin(b)theta and r=c+(a)sin(b)theta formulas with major tweaking were used to design this masterpiece. |
0 Comments
SSA is considered ambiguous because the provided information has the potential to create one, two, or zero triangles. This is because the side across from the given angle is able to be oriented in more than one way (see Figure 1). The Law of Cosines can't apply to SSA because there's no way of telling how the opposite side may be oriented. Figures 1 and 2 allow for a more in-depth description. Figure 1 displays possible orientations for when side a is greater than the height (h). Because there are two possible orientations, there are two possible triangles. If side a was equivalent to the height, only one triangle would be possible. Figure 2 shows what happens when side a is shorter than the height; there is no triangle possible. The different orientations for side a is what makes SSA ambiguous.
This project displayed how something simple like csc(x) could be "stretched out", and how a more complicated expression such as (1+secx)/(tanx+sinx) can be condensed. What's interesting about this is that there are many different ways to write a more complex version of cscx. That's actually where the challenge came from; deciding how far the equation on the left should be carried out. My project could have much simpler, or much harder. Creating a verification equation has many different possibilities.
This activity helped me see how sin, cos, and tan all affect the y values on the Cartesian plane. I think my biggest takeaway was learning that sin is rise, cos is run, and tan is rise over run. Up until this point, I didn't really understand the difference between something like sin(pi) and cos(pi). They're both the same angle on a circle graph, so they were essentially the same thing to me. Along with the special graphs, this activities allowed me to make the connection between the circle graph and the Cartesian plane. During the assessment without Mr. Cresswell, I was honestly winging it. The two special graphs were what enabled me to pass. Not knowing if i was right or wrong until the end was stressful to say the least.
With the exception of tangent and cotangent which equal pi , the periods of every graph is 2 pi. Amplitudes of the sin and cos are 2, and there is no applicable amplitude for the tangent, secant, cosecant, and cotangent graphs. The sine and cosine graphs show similarities, but the y-intercepts are different on each which would indicate that there is a horizontal shift. In fact, the cos and sin graphs could be identical if the c and d values were changed appropriately. The periods and amplitudes of the two graphs are the same, so you wouldn't need to add an a or b value. Tan, csc, sec, and cot have different vertical asymptotes at certain values because, for example, cot can also be defined as 1/tan x; when tan of x is 0, the cot ends up as an error. It results in an error because it equals 1/0, and you cannot divide by 0.
Part One
1. Stafford Loans: These are the most common type of loans, and basically anyone who applies is qualified. You're required to be a US Citizen, and cannot have any outstanding loans from the government. This loan has two categories: Subsidized (gov. will pay the interest while you're still in school, interest rate will also be a little lower) and unsubsized (you have to pay for the interest while in school, or you can deffer them until you graduate. There are several repayment plans once you begin to pay them off). - The current interest rates for Fed. Stafford loans are 4.66% for undergraduates, and 6.21% for graduated students. - Subsidized and unsubsized loans have the same interest rate, but subsized loans technically have a 0% interest rate while you're in school, making them cheaper overall. - These loans are compounded monthly, after the grace period following graduation. 2. PLUS Loans: These are for parents to pay for their child's education. These loans usually have a low fixed interest rate. Enrolling in an automatic payment plan grants a discount on the interest rate. The borrower can also deduct the loan from their taxable income, giving them a nice tax break. - Interest rates for PLUS Loans are 7.21% until July 2015. It's a fixed rate. - This loan is compounded monthly. 3. Perkins Loans: This is another federal student loan. It's similar to a PLUS loan, except that it's coming directly from the school. There are 1800 schools that use this method, and you must be attending one of them to qualify. This is geared towards those that are in serious financial need. This loan is a bit more selective about who qualifies due to its partnership with FAFSA. Naturally, the government makes the final decision here. - Interest rates for a Perkins Loan is 5.% for the duration of the 10-year payment period. - The Perkins Loan has a nine-month grace period. The payee starts paying in the tenth month, falls below part-time status, or drops out of school. - The compounding period for this specific loan is monthly. Part Two Assuming I take out $20,000 in loans over the course of 4 years, the most obvious option would be to take out a Stafford Subsidized loan. This option has a fairly low interest rate at 4.66%, and I won't have to worry about paying it off until six months after I graduate. Initial Loan Balance:$5,000.00 Loan Interest Rate:4.66% Loan Term:4 years Minimum Payment:$50.00 Deferment (Months):6 Capitalization Frequency:Monthly After the deferment period of 6 months, the new loan balance is $5,117.64 , including an additional $117.64. There will have to be 48 payments of $114.38 , for a total payment of $5,490.24 (including a total of $490.24 in interest) plus an additional $116.50 in interest paid during the deferment period. With the interest capitalization there are 48 payments of $117.07 , for a total payment of $5,619.36 (including a total of $501.72 in interest plus $117.64 in interest accrued during the deferment period). So the total amount paid with interest capitalization is $5,619.36 , or $12.62 more than would have been paid without capitalization. Sources http://www.debt.org/students/types-of-loans/ studentaid.ed.gov http://www.wiu.edu/vpas/business_services/loancalc.php The connection between factors and zeros is pretty simple, the zeros that you find from an equation is the factored function of that same equation. For example in the equation above, the zeros that were found are the numbers used in the factored form of the polynomial. Division helps us factor polynomials because it is a quicker way to find the solutions. The degree of the polynomial helps to predict the number of zeros by telling us in the highest degree, in the polynomial above the highest degree is 4, therefore that polynomial should have 4 zeros. Even though it does not look like it this polynomial has 4 zeros. There is one repeating zero in this equation which you would find out when using division (another reason division helps us factor polynomials). The highest power of the degree may not always tell us the number of factors because, like the example, there could be a repeated zero. Also it may not give the number of factors because it could be something that gives you an "ugly" square root or an imaginary solution. |
AuthorWrite something about yourself. No need to be fancy, just an overview. Archives
September 2015
Categories |