Graphing Sine and Cosine VISS Sum2025 Pre-Calculus Honor

7/24/25, 4:05 PM 6.1 Graphing Sine and Cosine: VISS Sum2025 Pre-Calculas Honors 6.1 Graphing Sine and Cosine In the following lessons we will examine techniques for sketching the graphs of the 6 trig functions. You have all the knowledge you need on the Unit Circle to graph these periodic functions. Let's get some vocabulary first. periodic function: any function that repeats a pattern on a particular interval amplitude: the height of the peak (or valley) from the midline midline: is the axis (or line) that the function oscillates, or "bounces" around period: the time interval in which the function repeats itself; how long it takes for the function to complete one cycle phase shift: the horizontal shift of the graph; i.e. left or right shift like we did with the transformations of other functions vertical shift: the distance the curve is moved up or down; the vertical shift of the midline In this lesson, we will develop the fundamental (parent) sine and cosine wave and then use with only amplitude and period change to transform the function. Then in the next lesson, we will begin to shift, or translate those functions. First let's understand the first three vocabulary words from the above list. 1/6 7/24/25, 4:05 PM 6.1 Graphing Sine and Cosine: VISS Sum2025 Pre-Calculas Honors and 0:00/4:45 Now let's try and "produce" a sine wave from the Unit Circle. Watch this video and notice how the triangle is formed and the angle rotates around the unit circle and the height of the triangle is dynamic (aka changing). The angle and the y value of that angle produces a graph of the function. The ordered pairs on the function are , which means we are graphing , aka . Sine Curve and the Unit Circle (Stationary Circle) 2/6 7/24/25, 4:05 PM 6.1 Graphing Sine and Cosine: VISS Sum2025 Pre-Calculas Honors Watch the video again. See the connection between the height of y and the function? We call this unwrapping the unit circle. Let's look more closely at the sine function. 0:00/4:17 What about the cosine wave? Well, it behaves similarly in that it is periodic and follows the unit circle. Cosine Function from Unit Circle So for you to sketch the graphs of sine and cosine, it is most advantageous to plot the five key points of one period: the intercepts, the maximum, and the minimum. Then connect those points with a SMOOTH curve. Repeat those 5 key points (critical points) for as many periods as you are directed to graph. 3/6 7/24/25, 4:05 PM 6.1 Graphing Sine and Cosine: VISS Sum2025 Pre-Calculas Honors So let me show you what I mean and give the general formula for sine and cosine. Then we will graph some problems. Note: When listing the critical points we only write the x coordinates... not the entire ordered pair. 0:00/9:03 And another, more complicated sine example... 10:31/10:48 How about a cosine problem... 4/6 7/24/25, 4:05 PM 6.1 Graphing Sine and Cosine: VISS Sum2025 Pre-Calculas Honors 4:21/6:42 Now, let's practice writing equations given a graph. For now, we will only write one equation. However, many correct equations exist for every graph. More on that to follow... 0:00/10:29 In order to be confident in our graphing work, you must practice and be organized. Soon we will do application type problems for periodic behavior. However, we can't do those until we can do the basic graphs. So let's practice. **Here is a very helpful resource and another "voice" of how to graph sine, cosine, and tangent. Don't worry about tangent yet. That's the next lesson. Just click along the numbers at the top of the window, just under the blue border. Graph sine and cosine (http://web.nmsu.edu/~kberver/HandGraphingTechniques/HandGraphingTechniques15.html) There are a plethora of online examples for graphing sine and cosine. However, I want the graph you show me to be accurate... NOT a sketch. 5/6 7/24/25, 4:05 PM 6.1 Graphing Sine and Cosine: VISS Sum2025 Pre-Calculas Honors When you graph sine and cosine I expect your answer to include the following: list all of the following: amplitude period x coordinates of critical points (there are five of them - intercept, max (or min), intercept at half period, min (or max), intercept at period) graph with the x-axis scaled off using the five critical points and the y-axis scaled to show the amplitude You do not need to use graph paper. If fact, I find graph paper to be more of a hindrance than a help. However, I do expect your axis to be straight and scaled. Just follow the examples as shown in my video. If you have a question about whether your problem is correct, just text me a pic and I will let you know. I welcome your questions and want to help you practice to perfection. This lesson is an excellent time to practice one of the most important mathematical standards: Attend to precision! Reminder: You are not to use MathWays or any other website to do your math assignments. If you do not graph sine and cosine in the manner demonstrated, showing the above components, you will not receive credit for your submissions. Please learn. There is much value in learning! Proceed to the next item to upload your notes. 6/6