Mr. Unit Circle

The unit circle has coordinates at various points but the main coordinates are (1,0), (0,1), (-1,0), and (0,-1). (1,0) is zero in radians and (0,1) is pi/2 in radians and (-1,0) is pi in radians and (0,-1) is 3pi/2 in radians. (Cosine, sine) is cosine for x and sine for y. The first quadrant is positive for cosine and sine, the second quadrant is negative for cosine and positive for sine, the third quadrant is positive for cosine and negative for sine, and the fourth quadrant is positive for cosine and negative for sine. The unit circle is helpful for figuring out if a triangle is 45, 45, 90 triangle or 30, 60, 90 triangle. The unit circle is also the simplest way of finding lengths and angles of triangles. All sine, cosine, and tangent points have similar cordinates. 

Law of Sines/ Cosines

The Law of Sines establishes a relationship between the angles and the side lengths of triangle ABC. In total there are three sines. Another important relationship between the side lengths and the angles of a triangle is expressed by the Law of Cosines. The expression itself involves a single cosine, but by rotation or symmetry similar formulas are valid for other angles. Also cosine unlike sine changes, it's sign in the range from 0 degrees to 180 degrees of valid angles of a triangle. Sine is always positive in this range, cosine is positive up to 90 degrees where it becomes zero and is negative afterwards.

Chapter Four Overview

 The sine definition basically says that, on a right triangle, the following measurements are related the measurement of one of the non-right angles, the length of the side opposite to that angle and the length of the triangle's hypotenuse. Alternately, the cosine definition basically says that, on a right triangle, the following measurements are related: the measurement of one of the non-right angles, the length of the side adjacent to that angle and the length of the triangle's hypotenuse. Therefore cosine equals adjacent over hypotenuse and sine equals opposite over hypotenuse. In chapter four we examine conditional trigonometric equations, that is, equations that are true for only certain values of the variable. When solving for the solutions for these equations there might be an infinite amount of answers because of the periodicity of the trigonometric functions. Also, to verify a trigonometric identity we use the fundamental trigonetric identities, the even-odd properties, and the basic arithmetic and algebraic operations. In order to verify a trigonometric identity, we are required to show that the given expressions are equivalent. 

Trigonometric Equations

In earlier sections we examined trigonometric identities, which are equations involving trigonometric functions that are satisfied by all values of the variable for which both sides of the equality are defined. In this section we examine conditional trigonometric equations, that is, equations that are true for only certain values of the variable. When solving for the solutions for these equations there might be an infinite amount of answers because of the periodicity of the trigonometric functions. In general, to obtain solutions of an equation such as sin x=radical 2/2, it is more convenient to use a unit circle and reference angles rather than a graph of the trigonometric function. When solving an equation, if you divide by an expression containing a variable, you may lose some solutions of the original equation. To prevent the loss of a solution you must determine the values that make the expression zero and check to see whether they are solutions of the original equation. 

Verifying Identities

A trigonometric identity is an equation or formula involving only trigonometric functions that is valid for all angles measured in degrees or radians or for real numbers for which both sides of the equality are defined. To verify a trigonometric identity we use the fundamental trigonetric identities, the even-odd properties, and the basic arithmetic and algebraic operations. In order to verify a trigonometric identity, we are required to show that the given expressions are equivalent. Some suggestions for verifying identities includes: simplifying the more complicated side of the equation first and finding least common denominators for sums or differences of fractions. Also if the two preceding suggestions fail, then express all trigonometric functions in terms of sines and cosines and try to simplify. In the end, don't treat a trigonometric equation as an identity until after you have proven that it is really true. 

Tangent Graph

The tangent function has a completely different shape, it goes between negative and positive infinity, crossing through 0. At pi/2 radians, or 90 degrees the function is officially undefined, because it could be positive infinity or negative infinity. For a tangent graph one cycle occurs between -pi/2 and pi/2. There are vertical asymptotes at each end of the cycle. The asymptote that occurs at pi/2 repeats every pi units. The period is pi and there is no amplitude because the graph continues on forever in vertical directions. 

Tangent graph:

Sine and Cosine Functions

For each real number t there corresponds an angle of t radians in standard position. We denote the point of interesection of the terminal side of the angle t with the unit circle by P(t). The x and y coordinates of this point give us the values of the six basic trigonometric functions. The y coordinate of P(t) is called the sine of t, while the x-coordinate of P(t) is called the cosine. The sine definition basically says that, on a right triangle, the following measurements are related the measurement of one of the non-right angles, the length of the side opposite to that angle and the length of the triangle's hypotenuse. Alternately, the cosine definition basically says that, on a right triangle, the following measurements are related: the measurement of one of the non-right angles, the length of the side adjacent to that angle and the length of the triangle's hypotenuse. Therefore cosine equals adjacent over hypotenuse and sine equals opposite over hypotenuse. 

Chapter 3 Summary

For Chapter three there are many different methods such as Approximate- midpoint, rational zero test, remainder theorem, long division, synthetic division, conjugate theorem, and number of zeroes which is degrees. When given a real zero in a problem there are three different steps to do: 1. Division (synthetic or long division), 2. Continue dividing, 3. Quadratic- either quadratic formula or factoring. When given imaginary zeroes in a problem there are also three different steps: 1. Write conjugate pairs as factors, 2. Multiply, and 3. divide. If your given nothing in a problem then you must use the rational zero test, test all possibilities and find one that works. When doing synthetic division remember to always switch the sign of the number you are using or else your answer will be wrong! Also, for the remainder theorem, the number you are given, you can substitute it into x. For the rational zero test, once you find a number that works, you must also find other zeroes that work too! 

Picture of me doing a problem involving long division: 

Rational Functions

Rational Functions are any functions which can be defined by a rational fraction for example an algebraic fraction such that both the numerator and the denominator are polynomials. A function  is called a rational function if and only if it can be written in the form

. Where  and  are polynomials in X and Q is not the zero polynomial. The domain of f is the set of all points X for which the denominator  is not zero. However, if P and Q have a non constant polynomial greatest common divisor R, then setting  and  produces a rational function . The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. Rational functions with degree 1 are called Mobius transformations and form the automorphisms group of the Riemann Sphere. Rational functions are representative examples of meromorphic functions. 

Graph of a rational function:

Zeroes of Functions

A zero of a function f is a number c for which f(c)=0. A zero c of a function f can be a real or a complex number. A complex number is a number of the form z=a+bi, where I squared equals -1 and a and b are real numbers. The number a is called the real part of z and b which is called the imaginary part of z. The symbol I is called the imaginary unit and it is common practice to write it as I=square root of -1. If z=a+bi is a complex number, then z=a-bi is called its conjugate. Thus the simple polynomial function f(x)=x squared plus 1 equals zero are plus/minus square root of negative one, that is , I and -I. 

Here is a picture of a problem that involves zereos of a function! 

Piecewise- Defined Functions

A function f may involve two or more expressions or formulas, with each formula defined on different parts of the domain of f. A function defined in this manner is called a piecewise- defined function. Piecewise functions are written using the common functional notation, where the main part of the function is an array of functions and associated subdomains. Most importantly, there must be a finite number of subdomains, each of which must be an interval, in order for the overall function to be called "piecewise". For absolute value piecewise functions all values of x less than zero, the first function (−x) is used. Which cancels the sign of the input value, making negative numbers positive. For all values of x greater than or equal to zero, the second function (x) is used, which evaluates trivially to the input value itself.

 

Here is a picture of a piecewise function being graphed. 

The F(x) Men- Translations and Transformations

Explanation:

In translations there are vertical and horitizontal shifts, depending on wether the number is inside parenthesis or outside the parenthesis. There is also a rigid transformation, which is a transformation that changes only the position of the graph in the xy- plane but not it's shape. One way of rigidly transforming a graph of a function is by a reflection in a coordinate axis. If y=f(x) is a function, then the graph of y=-f(x) is the graph of f reflected in the x-axis and y=f(-x) is the graph of f reflected in the y-axis. A nonrigid transformation is when the shape of the graph is changed but retains, roughly, it's original shape. Some examples of nonrigid transformations are stretching or compressing. A graph is vertically stretched by a factor of c units if c> 1, and vertically compressed by a factor of c units if 0<c<1. 

What is a function?

A function from a set X to a set Y is a rule of correspondence that assigns to each element x in X exactly one element y in Y. A function is usually denoted by a letter such as f, g, or h. The set X is called the domain of f. The set of corresponding elements y in the set Y is called the range of the function. The unique element y in the range that corresponds to a selected element x in the domain X is called the value of the function at x, or the image of x, and is written f(x). Since the value of y depends on the choice of x, y is called the dependent variable and x is called the independent variable. There are many types of functions such as linear, quadratic, piecewise-defined, combining and inverse functions. For example a linear function is f(x)=ax+b where a doesn't equal 0 and a and b are constants. 

What I learned this week in Mathland!

I have learned so many things in Mathland this week from real numbers all the way to the midpoint formula. The class went over what is a rational and non rational number and what's an integer again. We also learned inequalities again, understanding when to use an open and closed circle. I learned something totally new this week called the Sign Chart Method which helps to write an inequality an interval form. This week, we also went over the properties of absolute value, distance and midpoint formula. After solving an inequality, we learned how to graph it on a number line as well as drawing the point on a rectangular coordinate system. This week was a review for me, which helped me get back into my school mode.

Myself and things to know about me!

My name is Marissa Dickey and I am a junior at Maranatha High School. I play two sports: soccer and tennis and I even go skiing sometimes. I have played soccer for twelve years now and I am going to switch to a new sport which is tennis. I also have many hobbies such as cooking, reading, traveling and sailing. I love traveling and I hope to go to Europe one day and see the Vatican and all the famous sites surrounding it. One of my favorite places to travel is Canada because the weather is beautiful and it's not crowded like California. I am also Canadian and have many relatives there. Although my dream destination is probably Greece. Apart from my hobbies and sports, I have two dogs and only one sibling. My family is indeed very small.