Graphing Sine and Cosine Functions

Today we're going to take what we know about the unit circle and make that into a periodic graph. First we're going to make a table, then create a graph. Fill in the values for the sine of each angle from the unit circle. Use a decimal rounded to the nearest tenth if the value isn't a whole number. 0 0 π .7 π 1 3π .7 π 0 5π -.7 3π -1 7π -.7 2π 0 9π .7 Now take those points and plot them on the graph below, then connect the dots with a smooth curve. Based on what you see, make a guess about the negative x-values and graph those also (I won't grade the negative x-values, unless you don't do them. I just want to see what you think!) You've just created a graph of y=sin x! Sine, cosine, tangent, cosecant, secant, and cotangent are all called periodic functions. That means that they repeat (which makes sense, since when you travel around the unit circle, it repeats). Here is a bigger graph - try graphing more to the left and the right of the original graph. You can use the unit circle. Now we're going to look at the key features of this particular periodic function.Period: How often the function repeats. That means if you cut that section out of the graph, you could replicate it infinitely and get the whole graph. The period of y = sin x is 2π.Amplitude: How high up and and how far down the graph goes. You take half of the total height of the graph. The amplitude of y = sin x is1, since it goes up to 1 and down to -1. The rest of the key features are normal. They-interceptis0.There are infinitex-interceptsthat occur at0, π, 2π, 3π, ... Sometimes that is written askπwhere k is an integer.There are also infinitemaxima. They occur atπ 5π 9π can also be written asπ +2kπ, where k is an integer.And there are infinite minima. They occur at 3π 7π 11π can also be written as 3π +2kπ, where k is an integer. Here is the graph of y = cos x. Compare and contrast this graph and the sine graph. Write out pi (for example, write 2 pi or 3 pi If there are multiple answers, write the first three then 3 dots (for example, 0, pi, 2 pi, ...) Key Feature y = sin x y = cos x Period 2 pi 2 pi Amplitude 1 1 y-intercept 0 1 x-intercepts 0, pi, 2 pi, ... pi 3 pi 5 pi ... Maxima 1 1 Minima -1 -1 What do you think happens when you multiply sine or cosine by a number? Graph y = 2sin x, y = 1 x and y = sin x.In order to make the window correct, click on the wrench at the top right of the Desmos screen, then change it to this (the pi symbol is on the keyboard you can pull up at the bottom of the Desmos screen): Which key feature does multiplying by a number in front of the sine function change? Period Amplitude x-intercepts How does multiplying y = sin x by 2 affect the graph? Stretches it vertically Shrinks it vertically Stretches it horizontally Shrinks it horizontally Now match the graphs below with the equations that go with them. y = sin x y = 1.5 cos x y = 0.5 cos x y = 0.2 cos x y = 0.2 sin x y = 1.5 sin x