Polar Coordinates #4

In the coordinate (r, theta), r stands for the radius. If theta>0, then you rotate counterclockwise. If theta<0, then you rotate clockwise. When converting from polar (r, theta) to rectangular (x,y), substitute x=rcostheta and y=rsintheta. When converting from rectangular to polar, substitute r^2=x^2+y^2 and tan(theta)=y/x. For both of these conversions, solve for x or r. Trig identities are also key to this process.

Rotating Conic Sections #3

The starting point for rotating conic sections is to use the formula Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Identify all parts in order to being step 1. The first step is to find the angle using cot2(theta)=(A-C)/B. Step two is to replace x and y with either x=x'cos(theta)-y'sin(theta) or y=x'sin(theta)+y'cos(theta). The final step is to use algebra and simplify! A helpful hint is to use the different trig identities used previously so it will be easier to replace things. Rotating conic sections can be tricky, but the trickiest part is to not mess up on the basic algebra steps.

Parabolas #2

Parabolas are U shaped graphs. They have a focus, which is right above the vertex and that distance is called c and is (h,k+c) . The vertex is the center of the parabola such is (h,k). The directrix is y=k-c and is the line below the vertex. The parts described are for a graph that has a c>0, or positive. If the c<0, or negative, then the parts are reversed. The basic equation for a positive parabola is (x-h)^2=4c(y-k) and the basic equation for a negative parabola is (y-k)^2=4c(x-h). 

2nd Semester Goals

Three things I did well this semester were that I kept all my notes neat and color coordinated, studied hard and effectively (I hope) for the final, and asked for help when needed. Three things that I need to improve on are studying harder for all tests and quizzes complete all my WebAssign and blogs on time, and ask for more clarifying help. Last semester was probably the most difficult math time for me. Whenever I studied a lot for something and I think I understand it, the results didn't show it. I know I don't have the best study habits, but it is frustrating when I actually study, but don't get the grade I am aiming for. This semester, I am really going to try to push myself to ask more questions and get more help, even if I think I understand the concept.

This break, I went to San Diego with my basketball team. We took second place in our bracket and only lost by 2! 

Law of Sines and Cosines #15

To use the law of sines, a triangle must either have 2 sides and an opposite angle or 2 angles and an opposite side. The equations to use are sinA/a = sinB/b = sinC/c. When approached with the situation of sin being attached to a letter, take the inverse of both sides. A sin and its number are not allowed to be separated because they are one pair and can only be apart it the inverse is taken. The law of cosines is only used once, then use the law of sines. To use this law, a triangle must be given 2 sides and an included x or 3 sides. The formulas for the law of cosines are a^2=b^2+c^2-2bccosA, b^2=a^2+c^2-2accosA and c^2=a^2+b^2-2abcosC. 

Chapter 4 Summary #14

Chapter 4 was a difficult one. First, we learned about angles and their measurements. Standard position is when the initial side is in the positive x-axis. Coterminous angles are angles that have the same terminal side. For complementary angles, add 90 degrees or pi/2. For supplementary angles, add 180 degrees or pi. The arc length formula us s=rtheta. Theta must be in radians and there are no exceptions! Then, we learned about special triangles: 45. 45, 90 degrees and 30, 60, 90 degrees. Then, we went into verifying trig identities. Then, the sum and difference, double, and half angle equations, which I found to be the most difficult. Inverse trig functions were relatively simpler for me. I learned to always unrationalize the arc length. 

Mr. Unit Circle #13

The unit circle is extremely helpful. It provides the cosine and sine values of the important degrees. It also gives the degrees in radians. The unit circle is a circle with a radius of 1. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. There are also trig functions on the unit circle.