Class and Instructional Unit:

 

In Geometry, we had a unit on trigonometric functions.

 

Claim of Growth:

 

Throughout this year in math, I grew in my understanding and use of trigonometric functions.

 

Specific Learning Endeavor:

 

The trigonometric unit covered most of the basic functions including their inverses. For example, cosine, Law of Sines, Law of Cosines, etc. We mostly used these functions in determining missing angles of a triangle or possibly finding the area of that triangle. This unit only occurred in Geometry since these functions strongly pertain to this class.

 

Purpose, Process, and Products:

 

This endeavor includes many parts but the basic process of us learning these functions start with first learning about them. We, the sophomore class, completed an activity that included us finding the angle that was made by staring at the top of an object, like a flag pole, and measuring the distance we were from the object. After finding the attended angle, we had to figure out the height of object we were looking at using the angle and the distance we were from the object, which basically means we had to use Tangent. Tangent is when you have an angle and 1 side of the triangle that is not the hypotenuse, the longest leg on a triangle, and you need to find the missing length. Then later on in class we learned about Cosine and Sine. In these functions, you still use the given angle but you use the hypotenuse to find the missing length of the triangle.

Cosine is when you have to find the length that is adjacent, Δx, to the given angle and you put this missing length, as a variable, over the given measurement of the hypotenuse. Sine is almost the same except for trying to find the length adjacent to the given angle, you are trying to find the length that’s opposite from it while using the missing opposite length, Δy, over the measurement of the hypotenuse.

 

We practiced using these functions quite a bit and the products that were produce were many practice problems that were located in our notebooks. For instance, we had to find if a ramp would fit state regulations if it was going to be 3 ft. high and at an incline of 4.76°. Other products included: Chapter 4 and 5 Quiz 1 and Chapter 6 Test. The purpose of this unit was to help us understand these functions but also make connections to how we can use these functions in real life. Also, I want to state that these are only three main trigonometric functions and they are many more functions. These even include the inverse version of these functions ( and ).

 

Artifact 1:

 

The first artifact that demonstrates my growth in understanding trigonometric functions is Chapter 5 Quiz 1. In this artifact, we had to solve three simple problems that implemented the use of three different forms of functions. The first problem was that we had to find the measurement of variable x with the given information of 38° and the current length of the hypotenuse: 8. Given the information I know and that the variable, x, is adjacent to 38°, which means I must use the function Cosine. Using the formula, ,I found the value of x was 6.304. Another problem, was slightly different because I was given the length of the hypotenuse, 22, and the length of the other leg, which was 8. With this information I had to find the value of the angle, Ө° and since I was given the ratio instead of the angle, I had to do the inverse of the normal function because the other leg of the triangle (not counting the hypotenuse) was opposite of the missing angle. I went through the function and found that Ө°=21.324.

 

How Artifact 1 Exhibits Growth:

 

This artifact shows my growth by demonstrating how I had to implement Pythagorean Theorem to figure out question three on the quiz. In this artifact, I had to determine if all three sides of the right triangle made a Pythagorean Triple, which is when you go through the regular formula= and the answer of the missing length is a whole number. Before this time, I knew what Pythagorean Theorem was used to find the missing length of a right triangle when you are given 2 out of the 3 lengths but I didn’t understand how to determine if the sides of a right triangle formed a Pythagorean Triple. Now, I know that when you find the measurement of one of the missing sides, and if it’s a whole number, then that makes a Pythagorean Triple.

 

Artifact 2:

 

The second artifact that demonstrates my growth in understanding and use of trigonometric functions is the Chapter 6 Test. In this artifact, we had to solve many different triangle based problems, but four problems focused on the use of trigonometric functions. For example, one problem had me determine if the lengths of two different right triangles corresponded. I solved this problem by using Pythagorean Theorem on triangle ABC given that I have the measure of the hypotenuse, 14, and the measure of the other leg of the triangle, 7, which I found the measure of the missing length, line AB was. Then I had to use the theorem again for triangle DEF yet this time given the same measurement for the hypotenuse, 14, but the other length of the triangle was, which the measure of the missing length of the triangle, line EF, was 7. After finding the missing side lengths of the right triangles, I determine that the triangles lengths do correspond since line AB (on triangle ABC) was and this was the same for line DE (on triangle DEF) and line AC (on triangle ABC )was 14, which was the same for line (DF on triangle DEF). To reiterate, this is only one of the four problems that use these functions.

 

How Artifact 2 Exhibits Growth:

 

This artifact also demonstrates my growth but instead of using Pythagorean Theorem I used the Law of Sines. On question 3, I was given a triangle with two angles, 100° and 68°, and the measurement of one lengths of the triangle, which was 71m. With this information I had to find the value of one of the lengths of the triangle expressed as the variable: x. I first used the Triangle Sum Theorem, which states that the sum of the interior angles of a triangle will also equal to 180°. After finding the missing angle was 12°, I then had the means to use Law of Sines to find x. Law of Sines is basically when you set up a proportion to the degrees that is directly opposite to the side length and this can be expressed in this formula:. I then inputted the values into the equation and received 15.921m was the value of x. Before, this problem I had to grasp the concept that in order to use this function you must have at least two degrees and the measurement of one side length that is directly opposite from one of the degrees. In addition, I grew in my use of the Law of Sines by solving it like a common algebraic equation.  Thanks to my growth I was able to use my new understanding of the Law of Sines to find the value of x and receive full points on the question.

 

Impact of Growth:

 

I see this change of understanding the many trigonometric functions as an ever evolving one; meaning that it still room for improvement and I don’t know all of the functions, yet I’m still gaining more knowledge and implying that knowledge to these problems that involve many forms of trig. My growth in understanding, and using the trigonometric functions help me in the future by allowing me to comprehend complex problems in Trigonometry and in many forms of Physics. Finally, this growth can help me in my possible future career as an Aerospace Engineer since some concepts will need the use of Physics, Trigonometry, and Calculus and the more I recognize these functions; the stronger my skills will be when I reach that level of math and science.