The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x) include the domain, range, period, asymptotes and amplitudes. The domain of a cosine and sine function is all real numbers and the range is -1 to 1. The period is 2π, and the amplitude is 1. They have no asymptotes. The domain of tangent is all real numbers except for π2+kπ. The range is all real numbers and the period is π. Tan has no amplitude and has asymptotes when x= π2+kπ.

The domain of a secant function is all real numbers except for π2+kπ. The domain of a cosecant function is all real numbers except for kπ. The range of both is (-∞.-1]U[1,∞) and the period is 2π. Secant has asymptotes when x=π2+kπ. Cosecant has asymptotes when x=kπ. They have no amplitude. Cotangent’s domain is all real numbers except for kπ. The range is all real numbers and the period is π. It has no amplitude and has asymptotes when x=kπ.

In an inverse function, the x coordinate, or the domain, and the y coordinate, the range, switch places. Since only one to one functions have inverses, we take the interval -π2 to π2, which contains all the possible values of the sine function. Now, the new domain is [-π2, π2], while the range stays the same. We then switch the domain and the range, so the domain and range of arcsin (x) is [-1,1] and [-π2, π2]. For cosine, the interval [0,π] contains all possible values, and the range is still [-1,1]. To find arcos (x) we invert the domain and range again, to get [-1,1] as the domain and [0,π] as the range. For arctan (x), the interval (-π2, π2) includes all possible values. The range still remains all real numbers. Exchanging the domain and range gives us all real numbers as the domain and (-π2, π2) as the range.

As you can see, the properties of the six trig functions have many similarities and the inverse trig functions’ domain and range can be obtained with the one to one property of inverse functions...

...Tracy Gitonga
Modeling Data With Trigonometric Functions
Precalculus 1113-213
October 18, 2011
Real-life math is used in many activities that people do in a daily basis. In the next few paragraphs I will be explaining how to use a real world data and model it with a sine function of the form of y= a sin K (x-b). The graphs will use the law of sine which is defines as, “a law stating that the ratio of the sine of an arc of a spherical triangle to the sine of the opposite angle is the same for all three arcs.”
The following table gives the number of hours of daylight in Philadelphia, Pennsylvania.
Day | | Mar 21 | Apr 21 | May 21 | | June 21 | | July 21 | | Aug 21 | | Sept 21 | | Oct 21 | | Nov 21 | | Dec 21 |
Hours of Day-light | | 12 | 13.7 | 14.2 | | 14.8 | | 14.2 | | 13.7 | | 12 | | 11.4 | | 9.8 | | 9.2 |
While plotting the data, I realized that the sine function works better because the graph begins from the point (0) on the x-axis of the graph. Firstly, I realized I need to be able to find my equation before formulating a graph. My equation is y= a sin K (x-b). I need to find the “a”, which stands for amplitude, “the maximum extent of a vibration or oscillation, measured from the position of equilibrium.” I took the difference between the highest and lowest daylight hours reading in the table and divided by two.
14.8 – 9.2 = 5.6/2
A= 2.8
Secondly, I figured the...

...
PROPERTIES OF SINE AND COSINE FUNCTIONS:
1. The sine and cosine functions are both periodic with period 2π.
2. The sine function is odd function since it’s graph is symmetric with respect to the origin, while the cosine function is an even function since it’s graph is symmetric with respect to y axis.
3. The sine functions:
a. Increasing in the intervals[0, π/2]and [3π/2, 2π]; and
b. Decreasing in the interval [π/2, 3π/2],over a period of 2 π.
4. The cosine function is:
a. Increasing in the interval [π, 2π]; and
b. Decreasing in the interval [0, π], over a period 2π.
5. Both the sine and cosine functions are continuous functions.
6. The domain of the sine and cosine functions is the set of all real numbers from -1 to 1
7. The amplitude of both the sine and cosine functions is 1, since one-half of the sum of the lower bound is 1, that is ½[|1|]+[|-1|]=2/2 or 1.
8. The maximum and minimum values of the sine and cosine functions are 1 and -1 respectively, which occur alternately midway between the points where the functions is zero.
SINE FUNCTION COSINE FUNCTION
QUADRANT
AS S VARIES
VALUES OF SIN S
VALUES...

...Properties of Trigonometric Functions
The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x)are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points.
Sine Function: f(x) = sin (x)
* Graph
* Domain: all real numbers
* Range: [-1 , 1]
* Period = 2pi
* x-intercepts: x = k pi , where k is an integer.
* y-intercepts: y = 0
* Maximum points: (pi/2 + 2 k pi , 1) , where k is an integer.
* Minimum points: (3pi/2 + 2 k pi , -1) , where k is an integer.
* Symmetry: since sin(-x) = - sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0 , 0).
* Intervals of increase/decrease: over one period and from 0 to 2pi, sin (x) is increasing on the intervals (0, pi/2) and (3pi/2 , 2pi), and decreasing on the interval (pi/2 , 3pi/2).
Cosine Function : f(x) = cos (x)
* Graph
* Domain: all real numbers
* Range: [-1 , 1]
* Period = 2pi
* x intercepts: x = pi/2 + k pi , where k is an integer.
* y intercepts: y = 1
* maximum points: (2 k pi , 1) , where k is an integer.
* minimum points: (pi + 2 k pi , -1) , where k is an integer.
* symmetry: since cos(-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to...

...Properties and Functions of Ingredients in Baking
With simple ingredients such as flour, sugar, eggs, milk, butter, and flavorings a wide almost endless of products can be made. But to produce perfect quality products, careful attention must be paid to the ingredients in the recipe. Baking products depend on precise preparation. Baking is not an art. It is a science. It is important to follow baking formulas carefully and completely. “Different flours, fats, liquids, and sweeteners function differently. Bread flour and cake flour are not the same, nor are shortening and butter. If one ingredient is substituted for another the results can be different”. (Labensky)
There are many different types of flour. The most common flours are made from wheat but any grain can be used to make flour, like rice or corn. A grain of flour is made up of the bran, the endosperm, and the germ. The bran is the outer-shell of the grain. The bran adds texture and fiber to the flour. The bran also gives flours, such as whole wheat flours, their brown color. The endosperm is in the middle of the grain. Most simple whit flours use only the endosperm part of the grain. The endosperm contains a small amount of oil, carbohydrates, and protein. Gluten is found in the endosperm of the grain. However, gluten does not become gluten until moistened and manipulated, such as kneading. The germ is a concentrated source of nutrients located in the center of the grain....

...CIRCULAR FUNCTIONS
A different name of an angle is circular functions. Communicate the direction of a triangle to the length of the surface of a triangle. Trigonometric functions are important of triangles and form episodic occurrence, along with many complementary applications. Trigonometric functions have a wide range of uses including calculating indefinite lengths along with angles in triangles.
Trigonometricfunctions are normally specific as ratios of two sides of a right triangle including the angle, and able to equally specific as the lengths of different line segments from a unit circle.
More modern significance communicate them an infinite series or as solutions of specific different equations, allowing their extension to subjective positive also negative values and complex numbers. The sine with cosine functions are with usually used to model periodic function. Circular functions along angle θ are:
Sine Function:
sin θ = OppositeHypotenuse
Cosine Function:
cos θ = AdjacentHypotenuse
Tangent Function:
tan θ = OppositeAdjacent
Cosecant Function:
csc θ = HypotenuseOpposite = 1sinθ
Secant Function:
sec θ = HypotenuseAdjacent = 1cosθ
Cotangent Function:
cot θ = AdjacentOpposite = 1tanθ
A function of a direction in a right-angled triangle to be specific the...

...Trigonometric Functions Table
Function
Right Triangle Definition
Unit Circle Definition
Sine
Sine of theta is opposite over hypotenuse Sin θ =o/h
A unit circle is a circle with a radius of 1. In a unit circle sine of θ = y/r. r =1 so sin θ= y
Cosine
Cosine of theta is adjacent over hypotenuse Cos θ =a/h
A unit circle is a circle with a radius of 1. In a unit circle, cosine of θ = x/r. r = 1 sp cos θ = x
Tangent
Tangent of theta is opposite over adjacent Tan θ =o/a
A unit circle is a circle with a radius of 1. In a unit circle, tangent of θ = y/x. X cannot equal 0 because that is undefined
Cosecant
Cosecant is the reciprocal of sine. Cosecant, represented by abbreviation csc, is hypotenuse over opposite csc θ =h/o
In a unit circle, cosecant is the reciprocal of sine so sin θ = r/y. Y cannot equal 0 because that is undefined. So, csc θ = 1/y
Secant
Secant is the reciprocal of cosine. Secant is represented by the abbreviation sec, and is equal to hypotenuse over adjacent. Sec=h/a
In a unit circle, secant is the reciprocal of cosine so csc θ = r/x x cannot = 0 because that is undefined. So, sec θ = 1/x
Cotangent
Cotangent is the reciprocal of tangent, represented by the abbreviation cot. Cotangent of theta is equal to adjacent over opposite. Cot θ =a/h
In a unit circle, cotangent is the reciprocal of tangent so cot θ = x/y again, Y cannot = 0 because that would make it undefined.
...

...RUNNING HEAD: AQUARIUM DIFFERENCES 1
Freshwater vs. Saltwater Aquariums
C. Kevin Barr
ENG121
Andrea Bear
August 28, 2012
AQUARIUM DIFFERENCES 2
Freshwater vs. Saltwater Comparison
There are many differences when it comes to freshwater versus saltwater aquariums. We'll try and shed some light on some of the differences when it comes to tank types, aquarium setup costs, maintenance tasks associated with both tanks, and general cost of fish needed to maintain the aquariums. If you are thinking about this process than you will need to read this comparison since we have vast knowledge and experience converting these tanks. Please keep in mind that both setups allows the enjoyment and tranquility of the owner to watch the activity of the aquarium and lose themselves in the moment and forget about the daily stress of life.
In the freshwater environment there are different types of start ups that can be considered. These types of tanks range from natural setup, plain goldfish setup, planted tank setup, and different fish type setups. These types of aquariums do not vary that much in start up costs since there is not much variation in material needed for the start up. The saltwater world also has different tank types. These range from just having fish only in your tank, having fish with live rock, to having what is known as a reef...

...Section 5.2 Trigonometric Functions of Real Numbers
The Trigonometric Functions
EXAMPLE: Use the Table below to ﬁnd the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2
1
EXAMPLE: Use the Table below to ﬁnd the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2 Solution: (a) From the Table, we see that the terminal point determined by √ t = √ is P (1/2, 3/2). Since the coordinates are x = 1/2 and π/3 y = 3/2, we have √ √ π 3 3/2 √ π 1 π sin = cos = tan = = 3 3 2 3 2 3 1/2 √ √ π 3 2 3 π π 1/2 csc = = sec = 2 cot = √ 3 3 3 3 3 3/2 (b) The terminal point determined by π/2 is P (0, 1). So π π 1 π 0 π cos = 0 csc = = 1 cot = = 0 sin = 1 2 2 2 1 2 1 But tan π/2 and sec π/2 are undeﬁned because x = 0 appears in the denominator in each of their deﬁnitions. π . 4 Solution: √ From the Table above, we see that √ terminal point determined by t = π/4 is the √ √ P ( 2/2, 2/2). Since the coordinates are x = 2/2 and y = 2/2, we have √ √ √ π 2 2 2/2 π π sin = =1 cos = tan = √ 4 2 4 2 4 2/2 √ π √ π π √ 2/2 csc = 2 sec = 2 cot = √ =1 4 4 4 2/2 EXAMPLE: Find the six trigonometric functions of each given real number t =
2
Values of the Trigonometric Functions
EXAMPLE: π π (a) cos > 0, because the terminal point of t = is in Quadrant I. 3 3 (b) tan 4 > 0, because the terminal point of t = 4 is in Quadrant III. (c) If cos t < 0 and...