Performance Task- Letter to LCBC

Dear LCBC

It is an honor that you have given me the opportunity to propose to you my model of the unit circle. Trigonometry is a very important subject to consider, even in every day life. Trigonometry has many applications in the real world, as I will explain in my proposal. For now, I will explain how the trigonometric functions are derived. It all begins with a unit circle. A unit circle is a circle with radius 1. It is from this circle that we can see all 6 trigonometric functions. On this circle, draw a triangle, assuming a 30 degree angle.

We know that the hypotenuse has a length of one. To get the Sine of a triangle, you divide the opposite side by the hypotenuse. Because the hypotenuse is equal to 1, Sine is equal to the opposite side, which is shown by the red line. Cosine is found in a similar way. Cosine is the adjacent side divided by the hypotenuse. Again, since the hypotenuse is equal to 1, Cosine is equal to the adjacent side, as noted by the purple line. Shown below is the unit circle if we add Tangent and Secant. If you draw a line at the point (1,0) and extend it upwards to create a new triangle on the unit circle, it is easy to see where Tangent and Secant are found. First, let's derive Tangent. Tangent is the opposite angle divided by the adjacent one. Because this is a unit circle, we know that the adjacent angle is equal to 1, so the opposite angle is equal to the Tangent function, as shown by the color green. Secant is found by dividing the hypotenuse by the adjacent. We know that the adjacent angle is one, therefore, the hypotenuse is equal to secant, as shown by the color blue. Now we will build the third and final triangle in order to get Cotangent and Cosecant. This is done by extending a line out from the (0,1) position in the graph, as shown below. Just a note, that in this triangle, we will be using a different angle. Because the angle we have been using we have assumed is equal to 30 degrees, its complimentary angle must be equal to 60 degrees. From here we know that the other angle in the triangle is 30 degrees, which is the angle we will now be using. Cotangent is the reciprocal function of tangent, and is therefore found by dividing the adjacent angle by the opposite angle. From this diagram, we know that the opposite angle is equal to one. The adjacent side is equal Cotangent, as shown in pink.
By dividing the hypotenuse by the opposite side, we are left with Cosecant. And, since the opposite side we know to be equal to one, Cosecant is equal to the hypotenuse. And there we have all six trigonometric functions.


It is extremely important to have a strong grasp of the unit circle, trigonometric functions and what they all mean because trigonometry is very important to modern day society. My unit circle model and description clearly describe how the six trigonometric functions are derived, and during my proposal, I will describe the importance of these functions in modern day, common occupations. Thank you for your time.




Sincerely,




Student Name