Section 1-3
We first started this unit by proving the Pythagorean Theorem which is a2+b2=c2. We proved the theorem by completing this worksheet called Proof by Rugs. In this worksheet we used the habit of a mathematician where you break things into smaller pieces, we took a rectangle and split it up into multiple right triangles. From this we found the lengths known as “a”, “b”, and “c”. Then we went on to derive the distance formula using the Pythagorean Theorem. You may be wondering, what is the Distance Formula. Well the distance formula is √(x2−x1)2+(y2−y1). We constructed right triangles which was a much more direct proof from the Pythagorean Theorem . The next task we did was Using the Distance Formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane. We were able to complete this task by finding the distance between points on a Cartesian Coordinate Plane such as (4,3) and (-1,2). We were able to derive the distance formula by plugging in given points into x2+y2=4 which is the circle with a radius of 2 which was given to us, and solving for the distance between them.
Section 4-7
The next task we were tasked to do was defining the unit circle. The unit circle is basically a circle with a radius of 1 and has some points that end up being square roots and others being fractions. All these coordinates are obviously under one. The next task we were charged with was finding points on the unit circle (at 30 degrees, 45 degrees and 60 degrees). I mainly solved these problems with guess and check by plugging in numbers in x2+y2=1 since we were using the Unit Circle. The 6th task we were charged with was using the symmetry of a circle to find the remaining points on the unit circle. All we had to do was be familiar with a cartesian coordinate plane which is a four quadrant plane. Then next we would just apply our knowledge of each quadrants to all the points we had to fill out. The 7th task we were tasked to do was using the unit circle to define sine and cosine (of the angle theta). We did this by taking a triangle in the Unit Circle. Then from there we would try to find one of the 2 lengths of the 3 sided triangle, with all the angles given to us. From there we would use sine and cosine to solve the triangle.
Section 8-11
The 8th task Dr Drew gave us was trying to define the tangent function. I worked with what I had and what I remembered and knew from last year and came up with this definition as the tangent function; “The tangent function is a function you can put into a calculator where the opposite side of the *hypotenuse and the side adjacent to the hypotenuse are divided and “tangented”.
*hypotenuse
The longest side length of any triangle.
The next task we were instructed to do was using similarity and proportions to derive the general trigonometric functions (sine, cosine and tangent). We were able to derive this by working on a worksheet called Right Triangle Trig: Find the Missing Sides Lengths. We had to use the operations Sine, Cosine, and Tangent in order to complete the worksheet. The 10th task we were charged with was Using the unit circle to define the arcSine, arcCosine and arcTangent functions. These for me was a little tough since I didn't fully understand them when we were given the worksheet Right Triangle Trig: Finding Missing Sides and Right Triangle Trig: Finding Missing Angles . However as I progressed and used the habit of a mathematician by looking at the problem systematically by looking at them from a different angle, I was able to solve the problem and comprehend how it worked. The 11th task was using the Mount Everest problem to discover the Law of Sines ("taking apart"). We were given the background on Mount Everest and the information either the British or Europeans knew in order to find and pinpoint the location of Mount Everest. And with that we all attempted to solve for the location of Mount Everest or at least we thought we were. However I like most of my classmates didn’t use the Law of Sines which is the fastest way to get to finding the information in order to get the location of Mount Everest.
Section 12-13
The last two tasks we were to do and something we also learned was how to derive the law of sines and the law of cosines. We were able to learn and derive all this information and the laws of these operations by consulting to a worksheet called Trigonometry law practice where we were given information from a triangle and from there we figured out the law of sines and the law of cosines.
We first started this unit by proving the Pythagorean Theorem which is a2+b2=c2. We proved the theorem by completing this worksheet called Proof by Rugs. In this worksheet we used the habit of a mathematician where you break things into smaller pieces, we took a rectangle and split it up into multiple right triangles. From this we found the lengths known as “a”, “b”, and “c”. Then we went on to derive the distance formula using the Pythagorean Theorem. You may be wondering, what is the Distance Formula. Well the distance formula is √(x2−x1)2+(y2−y1). We constructed right triangles which was a much more direct proof from the Pythagorean Theorem . The next task we did was Using the Distance Formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane. We were able to complete this task by finding the distance between points on a Cartesian Coordinate Plane such as (4,3) and (-1,2). We were able to derive the distance formula by plugging in given points into x2+y2=4 which is the circle with a radius of 2 which was given to us, and solving for the distance between them.
Section 4-7
The next task we were tasked to do was defining the unit circle. The unit circle is basically a circle with a radius of 1 and has some points that end up being square roots and others being fractions. All these coordinates are obviously under one. The next task we were charged with was finding points on the unit circle (at 30 degrees, 45 degrees and 60 degrees). I mainly solved these problems with guess and check by plugging in numbers in x2+y2=1 since we were using the Unit Circle. The 6th task we were charged with was using the symmetry of a circle to find the remaining points on the unit circle. All we had to do was be familiar with a cartesian coordinate plane which is a four quadrant plane. Then next we would just apply our knowledge of each quadrants to all the points we had to fill out. The 7th task we were tasked to do was using the unit circle to define sine and cosine (of the angle theta). We did this by taking a triangle in the Unit Circle. Then from there we would try to find one of the 2 lengths of the 3 sided triangle, with all the angles given to us. From there we would use sine and cosine to solve the triangle.
Section 8-11
The 8th task Dr Drew gave us was trying to define the tangent function. I worked with what I had and what I remembered and knew from last year and came up with this definition as the tangent function; “The tangent function is a function you can put into a calculator where the opposite side of the *hypotenuse and the side adjacent to the hypotenuse are divided and “tangented”.
*hypotenuse
The longest side length of any triangle.
The next task we were instructed to do was using similarity and proportions to derive the general trigonometric functions (sine, cosine and tangent). We were able to derive this by working on a worksheet called Right Triangle Trig: Find the Missing Sides Lengths. We had to use the operations Sine, Cosine, and Tangent in order to complete the worksheet. The 10th task we were charged with was Using the unit circle to define the arcSine, arcCosine and arcTangent functions. These for me was a little tough since I didn't fully understand them when we were given the worksheet Right Triangle Trig: Finding Missing Sides and Right Triangle Trig: Finding Missing Angles . However as I progressed and used the habit of a mathematician by looking at the problem systematically by looking at them from a different angle, I was able to solve the problem and comprehend how it worked. The 11th task was using the Mount Everest problem to discover the Law of Sines ("taking apart"). We were given the background on Mount Everest and the information either the British or Europeans knew in order to find and pinpoint the location of Mount Everest. And with that we all attempted to solve for the location of Mount Everest or at least we thought we were. However I like most of my classmates didn’t use the Law of Sines which is the fastest way to get to finding the information in order to get the location of Mount Everest.
Section 12-13
The last two tasks we were to do and something we also learned was how to derive the law of sines and the law of cosines. We were able to learn and derive all this information and the laws of these operations by consulting to a worksheet called Trigonometry law practice where we were given information from a triangle and from there we figured out the law of sines and the law of cosines.