Chapter 4 Summary

In chapter 4, the main focus was on trig. We began by learning about the angles of the special triangles and their measurements meaning their values. Then we moved on to identifying the three fundamentals to trig: sine, cosine and tangent. We split them up so that sin and cos were in a group since tangent is so different from the other two. So we looked at the graphs of sin and cos and how they relate to the unit circle on this web simulator. Then together with the tan, we learned of the inverse functions of the sin, cos, and tan functions csc, sec, and cot. They were basically just reciprocate of eachother. Then we out the basics to use as we used sum and difference, half angels, and double angles along with inverses to solve questions.

Mr. Unit circle

Mr. Unit circle is our dear plate friend who got a lot of tattoos. On mr. Unit circle, there are the radians of the special triangle angles that are frequently used when loving for a trig equation or in trig in general. The circle is split up into halts, then thirds, quarters, and sixths on each hemisphere. The line that divides the entire plate in four are the axis, and they separate the quadrants that go from the top right to the left right to the bottom left to the bottom right as 1-4. The coordinates in radians is provided and so is the angle degree and value. 

Trig equations

There are three types of trig equations: sum and difference, half angles, and double angles. For each the sin, cos, and tan equations differ from one another. When using a sum and difference equation, the question will provide a ratio or degrees of sin, cos and tan. You want to split the given degree into workable part so that if you are given sin90, the. You would break it up into the angles of the special triangles to make sin(45+45). Same for the difference.

Otherwise, the equations are plug and chug and you just need to be able to identify which equation to use when. 

Piecewise functions.

A Piecewise function is defined as being made up of other functions. Usually there are two functions that make up a Piecewise function and they often come Ina linear format, although there are some exceptions. The two functions are then graphed separately onto the same graph depending on their domains. Two if the most well known Piecewise functions are that of absolute value and the floor function. The absolute value Piecewise functions is made up of the functions y=-x and y=x. The graph itself looks like a linear line from the origin that has been reflected onto both sides, therefore making a V or slanted right angle shape. The floor  function looks like a flight of stairs. The function hasnanninfinite number of pieces so it goes on forever. 

Zeroes of a function

The zero of a function is a number that can plugged in fr x in f(x) to solve the equation. There are two methods of finding the zeros - long division and synthetic division. Although the synthetic division is easier, it only applied to linear equations. For quadratics, the long division but be used. They long division is essentially the same as any long division problem except that there are variables dividing and being divided as well. Synthetics division is when you plug in the given zero and then write out the coefficients die each term and then multiply each product of each term by the divisor until you reach the last which is the remainder. The answer may be written as f(x)=d(x)q(x)+r(x). D stands for divisor, q for quotient, and r for remainder.

Law of sines and cosines

Law of sines is basically the rule that says that sinA/a=sinB/b=SinC/c therefor they are all equal to eachother. This is used If given 2 sides and an opposite angle or if given 2 angles and an opposite side. The law of cosines states that a^2=b^2+c^2-2(c)(b)cosA;  b^2=a^2+c^2-2(a)(c)cosB;  c^2=b^2+a^2-2(a)(b)cosA. This is used when given 3sides or she given 2sides and an included angle. However, the law of cosines should only be used once per question because it is easy to make mistakes.

Superhero transformations

Superhero transformations was an assignment that was supposed to help us remember and learn how to use the different functions of a line. There were seven different functions or superheroes that we used were linear, quadratic, power, ploynomial, rational, exponential, and logarithmic.a linear equation is a straight line and is identifies as f(x)=ax+b. A quadratic equation can be identified as y=x^2 and the graph looks like a U, otherwise known as a parabola.  Power function is exactly that, a function with a power and it's equation looks like f(x)=ax^b. An example of a polynomial function is y=x^2+x+b? A rational equation looks like y=2x+5/x-1. An exponential graph looks like a steep curve and is represented by a fixed base and a variable as an exponent unlike a power function. A logarithicmic function is y=log_2(x).

Verifying Trig Identities

Verifying, or proving, trig functions is only a matter Of plugging and chugging. There are some suggestions that you should follow though when verifying a trig function. 

1. Simplify the more complicated side: the equation will look something like sinx=sinxcosx. So the right side would be better to simplify because it is more complicated.

2. Find the common denominators: this is a relative rule that applies to almost all maths and should be common sense by now.

3. Change all trig functions in terms of sin and cos: sin and cos are far easier to work with than tan which is sin/cos anyways.

4. Use an identity: an identity is exactly that- an identity! It's very similar ton the transition property used in geometry where if A = B and  B= C, then A = C. 

Tangent

Tangent is the trig function is the relation of the opposite side over the adjacent side. This is seen in the TOA part of SOH CAH TOA. Unlike the sin and cos functions of trig which are fairly similar in image, the graph of a tan line is completely different. The period of tangent is pi instead do two pi because the middle of the function goes up and through the origin. The graph also has vertical asymptotes so that each line is distinguishable. 

The graph goes infinitely up and down and cross the x-axis every pi instead of two pi as clearly shown in the picture above. 

Sine and cosine

A trig function is the function of an angle and relate to the angles of a triangle to the length of its sides. Sine is the trigonometric function that is equal to the ratio of the side opposite of a given angle to the hypotenuse in a right triangle. Cosine is the trigonometric function that is equal to the ratio of the adjacent side to the hypotenuse of a given angle in a right triangle. The ratios can be easily remembered through the acronym SOH-CAH -TOA. The first letters of each group stands for the trig function (sin, cosine, or tangent). The two letters following represent the ratios. So Sin(Opposite/Hypotenuse)=SOH and so on. 

The graph of sine begins at the origin while the graph for cosine begins at 1 or your amplitude. 

Chapter 3 Summary

In chapter 3, we basically went over almost all of the functions. We learned his to dividen ploynomial functions and then finding the zeros and factors of polynomial functions. An important fact to note when looking for question regarding then factors/zeros of a polynomial function is that a "root" is the same and interchangeable with "zero". We the went over the real zeros of a polynomial function which is basically funding deriving the zeros from the factorization. We learned to approximate zeros and the moved on to rational functions which is any function that can be define by a rational fraction.

This is an example of a polynomial function. 

Rational functions

A rational function is any function which can be defined by a rational fraction where both the numerator and denominator are polynomials. To graph a rational function, you aphave to first find the asymptotes and the intercepts. Then from that, you have to plot the plots then graph. The vertical asymptote can be found by setting the denominator equal to zero and the solving for x. Once the asymptotes are drawn on the graph, it means that the function cannot touch that line so the graph would maneuver around it. The horizontal asymptote can be found by comparing the exponents of the variables. So the largest variable in the numerator is compared to that if the denominator by following a set of rules. If n<m, n being the numerator, then the x-axis is the horizontal aymptote. If it is the opposite, the there is no horizontal asymptote and instead you would find a slant asymptote. If n=m, then the horizontal asymptote is y=a/b. 

Zeros of a Function

The zero of a fucntion is an x-value that makes a function into zero. It can also be called a root of a function of f or an x-intercept. The zero of a function can be found  by solving the equation for x. Whatever the x value turns out to be will be the zero of the graph because at that point, if plugged into the function, it will cause the entire graph to be zero. Roots, zeros, and x-intercepts can all be real or complex. An example would be x^2=4. If one was to solve for x, then the square root of 4 would be 2, and that 2 would be the zero of the function.