Law of Sines and Cosines

The law of sine and cosine compares parts of a triangle. There are many formulas for the law of cosines, however the law of sines is a ratio, which compares sin(X)/x = sin(Y)/y. The law of cosines is more complicated.  The formulas for law of cosine are all variations of c^2 = a^2 + b^2 - 2abcos(C).
It is basically the pythagorean theorem, but it includes a value for an angle, which has the - 2abcos(C).
The other sides of a and b just switch around the variables. Both these laws are widely used in life appliactions, as they do not need to account for right triangles.

Mr. Unit Circle

The unit circle is a very useful tool. It has a radius of 1, and shows the x and y values along the circumference. Using special case angles, we can find the lengths and values of right triangles that have one of the special angles. Again, cosine and sine come into play, as cosine=x value, and sine=y value. We can use the acronym ASTC, all students take calculus, which helps remember which function is positive in which quadrant. A is all, S is sine, T is tangent, and C is cosine. We made our own unit circles in class using paper plates, and we drew all the angles and also radian measures on them. This was very useful on the test, and saved lots of time!

Chapter 4 Overview

Hi,

Today I will be talking about chapter 4. This chapter was very long, consisting of 10 lessons of trigonometry. However, it was very interesting, relating the sides of triangles to the unit circle, which accounts for special triangles and angles. The things to remember are (cos,sin) = (x,y), and cosine is equivalent to adjacent side over hypotenuse, whereas sine is opposite over hypotenuse and tangent is opposite over adjacent. This information allows us to find the points along the unit circle, finding the coordinates along the circumference using 45-45-90 and 30-60-90 right triangles sine and cosine values. Additionally, we learned that trigonometry has many identities, which are tools used to interpret and manipulate trigonometric equations for solving. We can prove the existence of these identities using verifying. Lastly, inverses are for solving for arcsin1/2 and we are solving for x. We solve by taking the sin^-1 of 1/2 and that is x.

Thanks,
Kaili Chiu

Trigonometric Functions

Hi,

Today I am going to talk about Trigonometric functions. These are dealing with lots and lots of triangle values. cosine, sine, and tangent, as well as their reciprocal functions are found using identities and a useful acronym SOHCAHTOA. When solving trig functions, one can perform many algebraic tricks and action such as factoring and the quadratic formula. If one was to solve say sinx=1/2, we could find a series of angles in 0 to 2pi. Overall, this is a very interesting way to look at triangles. The pythagorean theorem even factors into how the identities are derived!

Thanks,
Kaili Chiu

Verifying Identities

Hi,

Today, I am going to be talking about verifying identities. This is a very complex topic, as it is basically proving the identities in various situations of trigonometric functions. There are a bunch of tips and rules. First, you can only work on one side of the graph. Two, it is suggested that you pick the more complicated side, although when checking your work, you could use the other side. Another tip is to remember the algebra tricks such as grouping and multiplying by the conjugate. When you've mastered all of these tips and tricks, you will be able to verify!

Thanks,
Kaili Chiu

Tangent

Tangent is a very interesting operation. It is the brother of cosine and sine, and involves both of them in its composition. Sine over cosine equals tangent, or opposite over adjacent. As can be assumed from the previous blog post, tangent is found in right triangles. Even more interesting, the tangent graph has asymptotes. Its counterpart is cotangent, or 1/tan on a calculator or in a calculation. However, tangent is also a term used to mean touching at one point.

Sine and Cosine

Sine and cosine are two of the 6 important operations that one can perform on triangles. However, the only type of triangles one can use these two on is right triangles. Right triangles have one angle of 90 degrees, and using sine and cosine, we can find missing sides or angles. Sine and cosine on a calculator are represented by sin and cos. Sine is equal to the opposite side over the hypotenuse, whereas the cosine is equal to the adjacent side over the hypotenuse. Additionally, on a unit circle and also cartesian plane, cosine is equal to the x value, whereas the sine value is equal to the y value. We can use these to substituted x and y values and find specific lengths.

Chapter 3 Summary

Chapter 3 was all about functions. This involved rational functions, complex numbers, and factoring. We used many methods to solve and factor, like the foiling method, quadratic formula. Also, we found zeroes, which are the solutions of a function. An example of a quadratic function is f(x) = 4x^2 + 2x - 6.
The zeroes of this function can be found using the quadratic formula, or [-b +/- sqr(b^2 - 4ac)]/2a. When you plug in the values for the function, you get two solutions because of the +/- sign. This is where 4 = a, 2 = b, and -6 = c. Overall this was a very interesting chapter, although it is review.

Rational Functions

Hello,

Rational functions, the amazing nature of them. First, we learned about Rational functions in a test. This is called the Rational Zero Test, and is represented in p/s = factors of constant/factors of leading coefficient. The constant is the number at the end of function, and the leading coefficient is the coefficient of the biggest degree of the variable. When one of these works in f(x), you can use synthetic division to find the equation and eventually factors. Next, we used the midpoint equation to estimate a point to the utmost accuracy. Lastly, we understood rational functions and finding certain points and parts to a function. We found holes, asymptotes, x and y intercepts.

Thanks,
Kaili Chiu

Zeroes of Functions

Hello,

The zeroes of a function are known by many names. Some of these names include the x-intercept, x value. However, they all boil down to one thing. Solving for x. In a function f(x), if f(x) = x - 1, the zero would be 1, as when one solves for x, by adding a 1 to both sides, one can isolate the x. Zeroes are a key unit to solving any math equation, and are often asked for in many math problems inside and outside the classroom. Finding the zeroes is incorporated in algebra, calculus, math analysis, and geometry, and trigonometry. Pretty much every aspect of math.

Thanks,

Kaili Chiu

Piecewise Functions

Hi,

Today in Mathland, I learned about Piecewise functions. The funny thing about these is that they are multiple functions put together in the Cartesian Plane. We found the domain of the functions, and used it to plot each of them. Later, we discovered whether they were continuous or discontinuous. This can be found just by looking at the visuals of the plotted functions. If it is discontinuous, we would state where and at which points it was discontinuous. In addition, we also learned how to plot and identify [(x)] functions.

Thanks,
Kaili Chiu

Superhero Functions

Hi,

In the assignment for the f(x) men, we used specific functions to complete missions. This project allowed us to learn about transformations in functions, and how they are represented on the Cartesian plane. We used specific functions as superheroes to defeat the evildoers by transforming the function so that the function would hit certain points where the targets were. In certain graphs, we had to combine heroes and eliminate in the least amount of moves possible, and in others, we could only move along a certain axis. Some of these functions were 2^x, linear functions, parabolas, cubic functions, absolute value functions. The hardest mission was "The Outer Demon - Magic Foxes", where we had to finish the mission using negative functions. Overall, this was an interesting assignment that allowed us to understand transformations of functions.

Thanks,

Kaili Chiu  

    

What is a function?

Hi,

Today in the land of Math, I learned about functions. A function is an equation like f(x) = 2x + 1. F(x) is the f times x value, and can be used as the y for use in solving the equation. Domain is an important part of functions, and explain the x-entries possible. The y values are represented in range. Domain can be almost anything. In the case of square roots, however, one cannot put negative values in as x, since negative radicals are not in the real range. In a fraction, the denominator cannot be 0, or undefined.

Thanks for reading,
Kaili Chiu

What I Learned This Week

Hi,

I learned much over this week in Mathland even though I am still only a squire. I learned about Absolute values, circles in addition to Sir Cumference. Most of the concepts that were reviewed were easy and pretty easy concepts, although the review was needed, and refreshed my memory. In class we learned how to set up our web assign accounts and also troubleshooted problems with our Ipads. In the way of learning how to graph circles, we relearned completing the square in order to create a usable equation. In addition to circles, we also learned semicircles and absolute values. Absolute values are the distance from an integer to zero, and therefore all numbers' absolute values is positive, since there cannot be negative distance. Overall, it was an interesting week in Mathland.

Thanks,
Kaili Chiu

All About Me

Hi,

My name is Kaili Chiu. I am 15 years old, and my birthday is January 20, 1999. Throughout my early life (from 5-11) I was a happy child, and was very interested in the sport of Archery. I picked up the sport and progressed, shooting in competitions such as the Nationals. Throughout my 15 years, my parents, Lihu and Lucille have been caring, understanding, and patient, teaching me essential lessons and instilling morals. My sister of 13 years is very mature, and is often mistaken to be older than me, competitive, and artistically and musically gifted. When I was 4, I enrolled in High Point Academy, a Primary School from grades 1-8. At around the grade 6, I picked up the sport of Golf, and was intrigued by the complex nature of the activity. To this day, I still play golf for the Maranatha Golf Team, and am happy with my life at the moment. I enjoy other interests like art, music, and video games, although for most days out of the year, I attend a Christian school called Maranatha High School.

Thanks,
Kaili Chiu