Journal on "Pre-Calc Trigonometry"

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Modeling Real-World Data with Sinusoidal Functions

The sinusoid which is sometimes referred to as the sine wave referrers to a function of mathematics describing a smooth oscillation that is also repetitive. It usually takes place in pure mathematics and also in physics, electrical engineering and signal processing besides numerous other fields. Its form as a function of time (t) is:

A, which is the amplitude. It is the peak deviation of the function from its center position.

, which is the angular frequency, specifies the number of oscillations occurring in a unit time interval, in radians per second, which is the phase, gives specifications where in its cycle the oscillation begins at t = 0.

The swinging of an undamped spring-mass organization round the equilibrium is referred to as a sine wave. The sine wave is of great use in physics as it regains its wave shape when it is added to a different sine wave having similar frequency and also arbitrary phase. Sine wave is the only periodic waveform having this feature. This feature will result into its significance in the Fourier analysis and also makes it to be acoustically strange.

Generally, sinusoids are wave form graphs. Therefore, any phenomenon that has a periodic behavior or characteristics of a wave is capable of being modeled by sinusoids. This is including numerous simple actions like blood pressure in the heart, a pendulum, motion of an engine's piston-crankshaft, a child's swing, a Ferris wheel, tides, hours of daylight through out a year, (visible) shape of the moon, seasons, and sounds.

Examples of sinusoids are biorhythms

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of a person. The proponents of biorhythms are claiming that our every day lives are greatly influenced by the rhythmic cycles. These cycles are capable of interacting to make an indication of vigorous and inactive phases not only in the physical but also in emotional and mental aspects of humans. If one conducts a web search to establish biorhythm software, it will give out a series of waves. The people who use biorhythms do not state that biorhythms predict nor explain events. They however state that biorhythms recommend how we can cope up with them.

Another familiar instance of an action that is capable of being modeled by a sinusoid is the pendulum's motion. When we plot time against the angle that the arm of the pendulum makes with a vertical line indicating the location of the pendulum at rest will generate a sinusoid.

Cyclical behavior is also widespread in the business world. As there are periodic changes in the temperature of places, there are also seasonal changes in the demand for surfing equipment, snow shovels among several other things. The graph below gives a cyclical feature in employment at securities firms in the .U.S.

Law of sines

The law of sines which is also known as the sine formula or sine rule is applied in trigonometry. The equation tries to show the relationship between the lengths of the triangle's sides and the sines of the angles of the triangle. Given an example like this;

The law states that:

a, b, and c represents the lengths of the triangle's sides while a, B, and C. represents the opposite angles .on a number of situations, the reciprocal of this equation is used to state this equation. Thus;

The sine law is sometimes used in the computation of the remaining sides of the triangle given two angles and a side of the triangle. This technique is known as triangulation. Besides, it can be applied if two sides of the triangle and also one of angles that are not enclosed are given. In that given case, the formula can give two probable values for the angle that is enclosed. This always leads to an ambiguous case.

Law of cosines

The law of cosines is also referred to as the cosine formula or sometimes as cosine rule. It is a statement about a universal triangle that tries to relate the lengths of the sides of the triangle to the cosine of one of its angles. Using notation as in Using figure, the law of cosines states that represents the angle that is contained between sides of lengths a and b and opposite the side of length c.

The cosine rule generalizes Pythagorean Theorem. This theorem applies only to right triangles: when the angle ? is a right angle i.e. It measures 900 or ?/2 radians, cos (?) = 0, and therefore the law of cosines will reduce to;

The cosine rule is useful in the computation of the third side of a triangle if two sides and also their enclosed angle are given. Besides, it is used in the computation of the angles of a triangle when all the three sides of the triangle are given.

The following formulas also state the cosine rule.

Fibonacci numbers

Introduction

In mathematics, the Fibonacci numbers refers to the numbers that are in these given integer sequence;

Through definition, the beginning two Fibonacci numbers are 0 and 1. Besides, each succeeding number is the addition or the sum of the preceding two numbers. Other sources do not include the original 0, as an alternative, they begin the progression with two 1s.

Mathematically, the sequence Fn of Fibonacci numbers is always defined by the recurrence relation

Using seed values

The Fibonacci sequence has been named after Leonardo of Pisa. He was referred to as Fibonacci. Fibonacci 1202 volume which was called Liber Abaci made the introduction of the sequence to Western European mathematics (Laurence, 2002) despite that fact that the sequence was autonomously described in Indian mathematics (Goonatilake, 1998; Parmanand, 1985; Rachel, 2008 and Knuth, 2006).

Fibonacci numbers are widely applied in the examination of financial markets, in strategies like Fibonacci retracement. They are also widely applied in computer algorithms like the Fibonacci search method and also the Fibonacci pile data structure. The easy recursion of Fibonacci numbers has also stimulated a group of recursive graphs which are referred to as Fibonacci cubes and are used for interconnecting corresponding and also distributed systems. Besides, they come out in biological settings (Douady and Couder,1996) like the branching of trees, leaves arrangement on a stem, the spouts of fruits in a pineapple (Judy and Wilson,2006) flowering of artichoke, and the uncurling fern and also the arrangement of the pine cone (Brousseau,1969).

Fibonacci number patterns are always encountered. They occur so regularly in nature that we always get that the phenomenon is always called the "law of nature."

The petals of a pine cone always spiral in two directions. The number of petals going around once is usually a Fibonacci number. Seeds on a sunflower seeds are also showing the Fibonacci spiral. The patter is also found in pineapples.

Fibonacci sequence is shown by petals on many flowers

Number of Petals

Flower

3 petals lily and iris

5 petals buttercup, columbine vinca, larkspur and wild rose

8 petals

Delphinium and coreopsis

13 petals ragwort, cineraria and marigold

21 petals aster, chicory and black-eyed Susan

34 petals plantain, pyrethrum and daisy

55 petals

Daisy and the family of asteraceae

89 petals

Daisy and the family of asteraceae

Why these arrangements occur

Plants are always not aware of this sequence. They only grow in the most efficient ways. In the scenario of the leaf arrangement and phyllotaxis, a number of the scenarios might be related to maximizing each leaf's space and also the standard amount of light that falls on each of them. A tiny advantage would also come to take control over numerous generations (Grist, n.d)

Fibonacci sequence and population growth in animals

Fibonacci sequence copies the population growth pattern of animals. Beginning from one distinct offspring or 1 animal. After a period of one year, it matures up and is capable of reproducing .In a single year; it is capable of reproducing one offspring. Now they become 2 animals. In one single year, the mother will reproduce one fresh offspring and the offspring that is given birth to in the preceding year matures up. They now become 3 animals. In another year, the mother and the now mature offspring will each reproduce one offspring and the offspring that comes from the last year will become mature. There will be 5 animals now. The trend will go on. (Fuzzy, 2010)

How Fibonacci number works

During the year 1202, Fibonacci got interested in the reproduction of rabbits. He made an imaginary set of suitable conditions for rabbits to breed. He later posed the question, "How many rabbit pairs will there be after a year?" The ideal conditions that he set were as below;

1. You start with a single male rabbit and a single female rabbit. The rabbits have presently been born.

2. A single rabbit will attain sexual maturity after a month.

3. The period of gestation of a rabbit is a month.

4. Once a rabbit has attained sexual maturity, a female one will give birth on each month.

5. A female rabbit will usually give birth to… READ MORE

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Quoted Instructions for "Pre-Calc Trigonometry" Assignment:

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writing assignments and research assignments

short assignments

Writing assignment 1: bussiness application

Using Real World Application: Business on page 13 as a guide, research the admission fees and operating costs for a park, zoo, sports team, or other entertainment event near you.

Write and graph the linear relations that govern the fees and operating costs. Use this data to compare the cost of admission for your family or friends, to the operating costs of your chosen venue.

Compose these two linear functions to find a function that can be used to determine the profit. Predict the profit of the venue annually using these functions.

Writing assignment 2 Sports

Look up some data that changes every year, on your favorite sport. Cite your source(s). Examples are batting averages, pass completions, free throw percentages, etc. You should have approximately fifteen points to work with.

Graph the data and provide the following information:

1. Identify where the data is increasing and decreasing.

2. Label maximum and minimums as relative or absolute.

3. Select three critical points and demonstrate how to verify that they are the minimum, maximum, or point of inflection.

Writing assignment. terms and theorems

Create two tables listing and describing terms and theorems below. Illustrate each term or theorem with a graph.

Table 1

Terms:

Define each of the following terms in your own words:

Degree

Leading coefficient

Root

Illustrations:

Degree: show how this relates to the shape of a graph

Leading coefficient: show how this relates to the tails of a graph

Root: show how this relates to the x-intercepts of a graph

Table 2

Theorems:

State each of the following theorems in your own words:

Fundamental Theorem of Algebra

Factor Theorem

Rational Root Theorem

Illustrations:

Factor Theorem: illustrate with a polynomial of degree 3

Rational Root Theorem: illustrate with a polynomial of degree 4

The Location Principle: illustrate with a polynomial of degree 5

Research assignment temperature

Create a polynomial model for the low temperature each day in your town throughout the year.

Research the daily low temperature each day in your town.

Choose five temperatures spread across the four seasons.

Create a quartic model using regression.

Graph the points and regression model on the same graph.

Create a quartic model using matrices.

Graph the points and matrix model on the same graph.

Compare the two models: Are they identical or slightly different?

Do both models pass through all five points?

Here is the source for the model: http://www.geog.ucsb.edu/

Research Assignment: Shadows and Sun

Use a ruler, a clock and trigonometry to determine heights and times of day.

1) Record the length of the shadow of a foot-long ruler at 10 AM, at NOON, and at 2 PM. Use a level to make sure the ruler is vertical.

2) Use trigonometric ratios to determine the angle of elevation to the sun at 10 AM, NOON, and 2 PM.

3) Use your results to determine the height of a tree, a house, or a tall building. HINT: Measure its shadow at 10 AM, NOON, or 2 PM.

4) Use your results to determine the approximate time of day. HINT: Measure the length of the shadow of your ruler at some unknown time of day and compare that length to lengths of the shadows you found at 10 AM, NOON, and 2 PM.

5) Research the sundial. Write a one-paragraph explanation of how a sundial works.

6) Research the angle of elevation of the sun for your location and time of year. Compare the research data to your experimental data.

Writing Assignment: Laws and Cases

State the Laws of Sines and Cosines in your own words. Make a table summarizing the cases of how many triangle solutions exist.

1. Law of Sines: state the formulas, explain them in your own words, and draw a diagram illustrating one of them.

2. Law of Cosines: state the formulas, explain them in your own words, and draw a diagram illustrating one of them.

3. Create a table summarizing all the cases of how many triangles solutions exist. See Section 5.7, pages 320 and 321 of your text.

Writing Assignment: Modeling Real-World Data With Sinusoidal Functions

Average air temperature can be modulated by the sinusoidal equation t is equal to t sub average = t sub amplitude divided by 2 times sin of 2 times the quantity of t sub d plus 10 divided by 365 minue one over twelve minue pie over twelve.

Search the Web or physics texts to find three other sinusoidal equations that model real-world phenomena. Write a paragraph for each equation. For each example, address the following questions:

* What natural phenomenon does the equation model?

* What aspect of the natural phenomenon involves data that repeats after a given period of time?

* How could you use the equation to solve problems?

Research Assignment: Projectiles

Research the longest kick, throw, or home run of your favorite team.

1) Research the longest kick, throw or home run of your favorite team. If possible, try to find the angle and speed at impact. If you cannot find the angle and speed, try angles of 50Â°, 60Â°, and 70Â° and various speeds to come up with an estimate based on how far the ball traveled.

Perform the projectile analysis on the kick, throw, or home run.

2) Substitute Theta Imageand magnitude of vector v into magnitude vector v sub x equals magnitude vector v cosine theta and magnitude vector v sub y equals magnitude vector v sine theta to find magnitude of vector v sub x and magnitude of vector v sub y. Make sure to convert units if necessary.

3) Substitute your findings from step 2 into x equals t times magnitude vector v cosine theta and y equals t times magnitude vector v sine theta minus one half times g times t squared. Note that g = 32.

4) Solve t times magnitude vector v sine theta minus one half times g times t squared equals zero to find out long it took the ball to return to the ground.

5) Substitute one-half of the value of t you found in step 4 into y equals t times magnitude vector v sine theta minus one half times g times t squared to find the maximum height the ball obtained.

Research Assignment: Biography

Using Real World Applications, page 586 and 599 for guidance, research fractals of the form

f(z)= z2+ c.

Research one application and one image of fractals. Write a one-paragraph biography of key person in the development of fractals, other than Gaston Julia.

Choose a non-zero complex number c for f(z) = z2+ c. Find a complex number v that is in the escape set. Find a complex number w that is in the prisoner set. Graph the first five iterations of both v and w and connect them with line segments.

Challenge: Find a complex number j that is in the Julia set.

Writing Assignment: Modeling Real-World Data

The phrase *****garbage in, garbage out***** is often used to describe computer programs that work as planned, but provide inaccurate results because the input data is faulty.

The same concept can be applied to equations for modeling real-world situations. If the equations are based on data that is faulty or misleading, the models will produce inaccurate results.

Consider the exponential and logarithmic functions used to model real-world situations in Chapter 11-7 of your textbook. Look at the applications in both the Examples and Exercises sections. Choose three of the applications and write a paragraph for each one. For each of the applications, address the following questions:

* How do you think the data for this application was gathered?

* What problems do you think could arise when gathering data to model this type of situation?

* If data does not accurately reflect a real-world situation, what are the implications of using the resulting model to make predictions?

Writing Assignment: The Fibonacci Sequence

Many items and situations found in nature display the Fibonacci sequence. Examples include leaf-growth patterns, spirals in pineapples, and the population growth of some animals.

Find three examples of the Fibonacci sequence in nature. Write a paragraph for each example. For each example, address the following questions:

* How does the example relate to the Fibonacci sequence?

* What portions of each item or situation display the Fibonacci sequence?

* How could the Fibonacci sequence help you solve a problem involving the item or situation?

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