Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. Trigonometry is used a ton in the modern world. Some examples of real life applications include mapping, measuring the heights of buildings and objects, landscaping, and anything that uses angles and side lengths. It is also used in finding the distances of celestial bodies. Lets look at an example of using trigonometry to find the height of an object.
So lets say you have a building in front of you. You know that the building is 500 ft tall and you are 300 ft away from it. The angle from the point you are standing to the top of the building is37 degrees. We are using a special triangle for simplicity, but pretend you do not know the side lengths or the angles. Since it is a right triangle because the building goes up at a 90 degree angle, we can use SOCATOA. Since we know that a triangles angles should add up to 180, we can use subtraction to find the new angle. 90 + 37 =127. 180-127= 53. Using Pythagorean Theorem, we can find the last side length.
But what if the triangle is not a right triangle? What do we use then? There are number of ways in which we can solve non-right triangles. A few ways include the Law of Sines and Law of Cosines. Lets look at some examples of both:
To solve an SSA triangle
use The Law of Sines first to calculate one of the other two angles;
then use the three angles add to 180° to find the other angle;
finally use The Law of Sines again to find the unknown side.
To solve an SSS triangle:
use The Law of Cosines first to calculate one of the angles.
then use The Law of Cosines again to find another angle.
and finally use angles of a triangle add to 180° to find the last angle.
A picture demonstration:
Once again, these processes can be used in many different areas. Construction, art, mathematics, architecture, and many others including astronomy, surveyors, and pretty much anything using any type of triangle.
SSA is ambiguous because two different triangles can be made from the given information. You can't apply Law Of Sines simply because it only helps you solve one side length. I will help you find out if there is a solution though.
If you know the length of two sides and an angle other than the angle between those sides, then the Law of Sines can be used. This is the "SSA" case -- Side, Side, Angle. Assuming you know the lengths of sides a and b, and angle A,
a/sin(A) = b/sin(B) sin(B) = (b/a) sin(A)
If a < b, and sin(B) = (b/a) sin(A) is between 0 and 1, then two different angles, B, can satisfy this equation: one is acute, the other is obtuse, and these two angles are supplementary.
I didn't learn anything new, but I did figure out that the x and y on the unit circle are very important. The special triangles are super important too. Its good to know them and see them in the unit circle. Without the teachers help, it was a bit difficult. It was strange not being able to ask for help. That is good though, because it helps us remember and learn the material.
A radian is a type of measurement used to measure the arc length of a circle and many other shapes. It is basically a way to write the degrees in a more algebraically sound way. That way, we can calculate a lot more with the radian measurements. The unit circle gives the basis for all radians. It is the "Perfect" circle. 2pir relates to radians because the circumference is made up of 2pi radians. Using this and the radius length, we can find the circumference. Radians are like degrees but measure the An angle's measurement in radians is numerically equal to the length of a corresponding arc of a unit circle, so one radian is just under 57.3 degrees
The Fibonacci Sequence are the numbers in the following integer sequence: 1,1,2,3,5,8,13,21,34,55,89,144,... By definition, the first two numbers in the Fibonacci sequence are 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. The Fibonacci Sequence was first introduced by Fibonacci in his book titled, "Liber Abaci" in 1202. This book introduced the Fibonacci Sequence to western European mathematics. The Fibonacci Sequence is used in many computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. The Fibonacci Sequence also appears in nature often. It appears in the branches of trees, the leafs on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and many other examples. This sequence is very interesting because it occurs in nature all the time in many different forms, and we don't know why yet. Without the Fibonacci Sequence, we would not have computers or the internet!
The picture above shows the relation between Pascal's Triangle and Fibonacci's sequence.
The Golden Ratio can also be found using the Fibonacci Sequence.
There are three major types of loans given to students looking to further their education through college. The three types are subsidized, non-subsidized, and bank loans. Direct subsidized loans are available to undergraduate students who qualify for financial aid. Your school determines the amount you can borrow, and the amount may not exceed your financial need. The interest is paid for by the Department Of Education while in school and six months afterwards. If you defer your loans, the Department Of Education will pay your interest. If you receive a Direct subsidized loan that is first disbursed between July 1, 2012, and July 1, 2014, you will be responsible for paying any interest that accrues during your grace period. Non-subsidized loans are available to undergraduate and graduate students and there is no need to demonstrate financial aid. Your school determines the amount you can borrow based on your cost of attendance and other financial aid you receive. You are responsible for paying the interest during all periods. If you choose not to pay the interest while you are in school and during grace periods and deferment or "forbearance"periods, your interest will accrue (accumulate) and be capitalized (that is, your interest will be added to the principal amount of your loan). Bank loans can be acquired by any individual looking for monetary assistance with college payments including dorm, food, books, and more. Payment is not required until 6 months after leaving school. Having a cosigner will greatly increase your chances of receiving a loan and may help lower interest rates. Fixed interest rates for student bank loans range from 6.39% to 10.93%. Variable interest rates range from 3.17% to 8.60% for student bank loans. Subsidized loan interest rates average 4.66%, while non-subsidized loans average 4.66% also. All loans are compounded monthly unless told otherwise.
Subsidized loan: $5,000 for 4 years with 4.66 interest. 3.5 years of interest. (grace period)
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year
Example:
An amount of $5,000 has an annual interest rate of 4.66%, compounded monthly. What is the balance after 3.5 years?
Solution:
Using the compound interest formula, we have that P = 5000, r = 4.66/100 = 0.0466, n = 12, t = 3.5, Therefore,
So, the balance after 1 years is approximately $5883.91. $5883.91 * 2.5 = $36774.44 total amount owed after 4 years.