Law of Sine and Cosines

When learning how to use trigonometry to solve oblique triangles, it is most important to know when and how to use these two laws. If that’s enough for you, then just skip on to the next section on area of a triangle. But if you’re interested in why they’re true, then continue on. As usual, we’ll use a standard notation for the angles and sides of a triangle. 

The Law of Sines

 That means the side a is opposite the angle A, the side b is opposite the angle B, and the side c is opposite the angle C.

The law of cosines

There are two other versions of the law of cosines: a2 = b2 + c2 – 2bc cos Aandb2 = a2 + c2 – 2ac cos B.Since the three verions differ only in the labelling of the triangle, it is enough to verify one just one of them. We’ll consider the version stated first.In order to see why these laws are valid, we’ll have to look at three cases. For case 1, we’ll take the angle C to be obtuse. In case 2, angle C will be a right angle. In case 3, angle C will be acute.

Math Joke of the Week: 

Q: What is the definition of a polar bear?
A: A rectangular bear after a coordinate transformation

Unit Circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.If (x, y) is a point on the unit circle, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equationx^2 + y^2 = 1.

Math Joke of the week:

Q: Why didn't sin and tan go to the party?
A: Just cos

Chapter 4 summary

 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. Also, we learned that trigonometry has many identities, which are tools used for trigonometric equations to solve. We can prove the existence of these identities using verifying. 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.

Math Joke of the Week:

Q: What do you get when you cross a mosquito with a mountain climber?
A: Nothing. You can't cross a vector and a scalar.

Trigonometric Equations

Solving trig equations use both the reference angles you've memorized and a lot of the algebra learned.

Solutions of trigonometric equations may also be found by examining the sign of the trig value and determining the proper quadrant(s) for that value. 

Be prepared to solve sin(x) + 2 = 3 for 0° < x < 360°

Just as with linear equations, I'll first isolate the variable-containing term:   sin(x) + 2 = 3   sin(x) = 1

Now use the reference angles memorized:   x = 90°

Solve tan2(x) + 3 = 0 for 0° < x < 360°

There's the temptation to quickly recall that the tangent of 60° involves the square root of 3 and slap down an answer, but this equation doesn't actually have a solution:tan2(x) = –3

How can the square of a trig function evaluate to a negative number? It can't, no solution

Math Joke of the week:

Q: What did one Calculus book say to the other?
A: Don't bother me I've got my own problems!

Solving Trig Identities 

Proving an identity is very different in concept from solving an equation. Though you'll use many of the same techniques, they are not the same, and the differences are what can cause you problems.

Always start off with the bigger or harder side of the identities.An "identity" is an equation or statement that is always true, no matter what. For instance, sin(x) = 1/csc(x) is an identity. To "prove" an identity, you have to use logical steps to show that one side of the equation can be transformed into the other side of the equation. You do not plug values into the identity to "prove" anything. Identities such as: sin2q+cos2q=1 

tan2 q +1 = sec2 q 

1+cot2q =csc2q

are needed to use to solve the identities 

 

Math Joke of the week

Q: Why is beer never served at a math party?
A: Because you can't drink and derive.

Tangent
Tangent is opposite over hypotenuse. But using the using the caressing plane, it would be another branch to sine and cosine. Tangent is a very interesting operation. Tangent is found also in right triangles. But the tangent graph has asymptotes and 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.
 To find the x intercepts of a tangent graph, use the equation N(π)

Math Joke of the week

Newlyweds A newlywed husband is discouraged by his wife's obsession with mathematics. Afraid of being second fiddle to her profession, he finally confronts her: "Do you love math more than me?" "Of course not, dear - I love you much more!" Happy, although sceptical, he challenges her: "Well, then prove it!" Pondering a bit, she responds: "Ok... Let epsilon be greater than zero..." 

Sine and Cosine

Sine, Cosine and Tangent are all based on a Right-Angled Triangle.

Before getting stuck into the functions, it helps to give a name to each side of a right triangle: triangle showing Opposite, Adjacent and Hypotenuse

"Opposite" is opposite to the angle θ"

Adjacent" is adjacent (next to) to the angle θ

"Hypotenuse" is the long one.

For a triangle with an angle θ, they are calculated this way:

Sine Function:sin(θ) = Opposite / Hypotenuse

Cosine Function:cos(θ) = Adjacent / Hypotenuse

It is important to memorize 45, 45, 90 triangles and their corresponding sides and well as the sides of 30, 60, 90 triangles.

Math Joke of the week 

Applying For A Job 

There are three people applying for the same job. One is a mathematician, one a statistician, and one an accountant. The interviewing committee first calls in the mathematician. They say "we have only one question. What is 500 plus 500?" The mathematician, without hesitation, says "1000." The committee sends him out and calls in the statistician. When the statistician comes in, they ask the same question. The statistician ponders the question for a moment, and then answers "1000... I'm 95% confident." He is then also thanked for his time and sent on his way. When the accountant enters the room, he is asked the same question: "what is 500 plus 500?" The accountant replies, "what would you like it to be?" They hire the accountant. 

Ch. 3 Summary :'(

So basically in the last month, we just been trying to solve functions. We factored, used complex numbers, and solved rational functions. Chapter 3 involves dividing polynomials in order to find zeros, the graph of a polynomial depends on its end behavior. Even degree polynomials with a positive coefficients go up but negative coefficients it goes down. Odd degree polynomials with a positive leading coefficient go down then up and if there is a negative coefficient it goes up then down. Using long division and synthetic division, you can find factors which consist of the zeros of the function. Also by plugging a number into a function, you can see if it is a zero by looking at the remainder. If the remainder is 0 then the number is a zero. By using the P/S equation you can find numbers that may or may not be zeros of a function. Check by using dividend and the remainder will determine if the number is a zero. The last part of the chapter involves finding asymptotes through methods such as factoring the denominator of a function, dividing leading coefficients.

Joke of the Week: 

I just finished reading Newton's Principia Mathematica, and found much of it to be rather derivative.

Rational Functions

All of these "rational" function have asymptotes. They are rational because they know not to reach certain places. 

A rational function is any function which can be defined by a rational fraction. For example, an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials do not need to be rational numbers, they may be taken in any field K.

The asymptotes are represented into dotted lines. The curved lines would be the functions. Note, the functions are not tangent to the asymptotes but are very close as it reach the end.

Joke of the week: 

The mathematician worked at home because he only functioned in his domain.

Zeroes of a Function 

Whenever a function hits the x-axis, that point is known as a zero. Haha, how can a number greater than zero be a zero. 

When the function is set to be an equation, it is always set to zero. For example, x+4=0

Always isolate the x variable as it is where the point of which it meets the x-axis. 

It can be also called roots, since the x-axis is horizontal and respects the ground floor.

When you solve for a zero, it would be basic algebra. So if you haven't taken Algebra 1, you can be in big trouble because many functions are required to be solve using this method. 

Pun of the Week: 

Some mathematicians are reluctant to cosine a loan.

Peace wise Functions

1. "In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub functions. Each sub function applying to a certain interval of the main function's domain (a sub-domain)."

   I don't even know what that means. 

   

   From the looks of this graph, and from what Ms. V taught me was that the peace wise function ends with the function then continues. Each function will graph differently due to the restrictions. The middle function is usually one point to another. While the other two functions are continuos from one end to another. 

Joke of the Week

The fattest knight at King Arthur's round table was Sir Cumference. He acquired his size from too much pi.

This week got this weird project on how transformation are needed to reach certain point on an axis. Each reflection is turned into a "hero" identity and each have their own personality. Each problem is also given their story and how they must be solved. 

There are three main transformation: Reflection, rotation, ad translation 

The other important transformation is resizing (also called dilation, contraction, compression, enlargement or even expansion). 

Reflection: Every point is the same distance from the central line! and the reflection has the same size as the original image. The central line is called the Mirror Line.

Rotation: The distance from the center to any point on the shape stays the same.Every point makes a circle around the center.

Translation: Every point of the shape must move:

1. The same distance 

2. In the same direction

We got a special assignment but my work was required to be scanned and they are right below:

What is a Function?

My definition for function: each input will be given an output or answer

The internet's definition: A rule that relates two variables, typically x and y,  is called a function  if to each value of x the rule assigns one and only one value of y. When that is the case, we say that y is a function of x.

Miss V's definition: f(x)=a

Every function is given a separate equation and a different answer.

The x-values is defined as the domain, while the y-values is the range.

If fraction is given as the answer while a zero is the denominator, it considered undefined.

Also if the a negative number cannot be within a radical sign/square root as a negative number is not part of a range.

this week...... 

nothing

Just Kidding Ms. V!!!! :P

1. I learned about the real line and solving the equations

2. Set notation and interval notation

3. Sign Chart Method: 

4. That unbound intervals include ∞(brackets []) .

5. Absolute value:       x, if> 0

                             /x/=-x if < 0

6. Equation for a circle: (x-h)^2 + (y-k)^2 = r^2

                                    center=(h,k)

                                    radius=r

7. 

All About Me

• I am a big time of gaming

Social Media helps me so much, homework, stay connected to my friends, and the occasional funny videos and memes.

• I have pride in being of Asian heritage

• I enjoy being at Maranatha, I imagine about the others schools and I thought, "This school is like my middle school, way better than most school!"  
  
   
• My favorite artist/ rapper is J. Cole
  
• My favorite fruit is the lychee
  
• Favorite verse: Luke 10:27 and other versions of it