# Trigonometric Functions

As high schoolers go their their teenage years of high school they learn from a variety of subjects. From math to science to history, there is a depth of knowledge to be learned. For math 3 and math 4, we are introduced to the world of trigonometry. So far, we have learned that there are currently three main trigonometric functions, cosine, sine, and tangent. But today I want to explore the other side of the trigonometric world, the inverse functions of cosine, sine, and tangent.

For the first inverse function, cosine, we know that it is obviously the inverse of cosine. The cosine function uses a angle of a right triangle and gives use the ratio of the x-value and the r-value (“Trig Functions:

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But instead of tracking the x-value of our unit circle, shown in Figure 1, we track the y-value. This is because the function of sine used the angle of the right triangle but instead of giving us the ratio of the x-value and r-value, it gives us the y-value and r-value (“Trig Functions: Overview”). When we graph the parent function of sine, which is sinx, we get the graph in Figure 4. It look the same as cosine but is shifted the right by 90 degrees. So our roots will be -360, -180, 0, 180, 360 and so on. However, our range is at [1,-1] the same as cosine. For the inverse function of sine, it is the same scenario, shown in Figure 5. The graph is just shifted up 90 degrees from the inverse function of cosine. The x limit is the same scenario too. The period for this function is just different because the distance from the vertex are at a different

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For this function I want to find a way to be able to graph functions or inverse functions that is technically not a function, so there will be multiple y values for a x value. Constantly, teachers ask students to graph the inverse functions with and without a x-limit. In the more advance concepts of math, students need to have the ability to graph these inverse functions quickly. To do this, I am going to contact the company that makes the most popular graphing calculators, the Texas Instruments Company. I am going to propose them my idea for the ability to reprogram their newest graphing calculator, the TI-84. Then I am going to contact a programing company, Manta, to have them reprogram the calculator for what I want it to do. But first I have to find a way to graph these functions. For this I want to have a new interface for the inverse functions, just like the interface for the functions on the TI-84. To graph the function on the calculator, I want to be able to take the function and make it into the inverse function graph by switch the x-value and y-value at every point. This is because this is what algebraically happens between functions and its inverse function. Then I want it to graph the new coordinates it received, so it will show the user the inverse function. By doing this process and ignoring the function algebraically, I will get a inverse function without a x-limit,