Surprise! This week's fun thing is worth extra credit! I'm trying to see if anyone even looks at these, so if you do, here's an opportunity for 1 (1!) point of extra credit:
Do something nice for someone you're quarantined with. It can be anything. To get the extra credit, tell me what you did in the last field of the google forms activity for the week (in the "Ask a question about these topics (if you have one), or please let me know if you are struggling with anything." box).
Click here for your weekly checklist.
There are only slight procedural differences when it comes to graphing each trig function. The main reason we started with sine is that the relationship between the "y-value" around the unit circle and the central angle is probably the most intuitive. The relationship between the "x-value" on the unit circle and the central angle is how we go about generating the cosine curve. You'll see in the following video (and the .gif below) that sine and cosine are extremely closely related. In fact, the "co" in front of cosine, means co-function. Cosine is the co-function of sine. A useful property of trigonometric co-functions is that, for example, the sine of angle is equal to the cosine of the complementary angle (recall: complementary angles add up to 90° (or π/2)). [ ex: sin(35°)=cos(55°) ]
As a function, cosine is extremely similar to sine. Along the horizontal axis we are plotting angles as inputs, and outputs are the outputs of f(x)=cos(x), which we know are the x-values of different coordinate points along the unit circle.
Using all the same techniques we used for graphing sine functions, we can also graph cosine functions. Here are some examples:
Practice problems for graphing cosine functions: PG 134 #15, 24, 40, 41, 45, 46, 62, 69 (state the amplitude, period, phase shift, and vertical shift of each cosine function as you set up your graphs)
Sine and cosine have very similar patterns. For a positive sine function, the first dot will be placed wherever (0,0) was translated to (due to a phase shift and/or vertical shift). From there it's just a matter of determining how far apart each "tick mark" along the x-axis will be, and then laying down the pattern of zero, maximum, zero, minimum, zero. Similarly for positive cosine, we find where the point (0,1) was translated to, then lay down the pattern of maximum, zero, minimum, zero, maximum at the appropriate tick marks. The negative versions of sine and cosine would follow opposite patterns.
Though it looks quite a lot different from sine and cosine, tangent is still a circular (or periodic function). We can look at the values of tangent as we go around the unit circle to make some observations about its fundamental period and where it has domain restrictions.
Practice problems for tangent: PG 120 #17, 19, 21, 71, 77, 78, 85. 86, 87, 91, 92, 93.
When we first discussed tangent it was as the ratio of the length of the opposite side of a right triangle with respect to one of the acute angles and the length of the adjacent side of the same angle. When we went a step further and considered a unit circle, it became clear that the opposite side of every angle would just be the "y-value" of the coordinate point that falls at the end of the terminal side of the angle, and the adjacent side was the "x-value". Since the y-value is the sin(θ) and the x-value is the cos(θ), we just say that tan(θ)=sin(θ)/cos(θ). Because cosine is in the denominator, and cos(θ) is zero at odd multiples of π/2 (1π/2, 3π/2, 5π/2, etc.), we will end up domain restrictions to tan(θ) in the form of infinite discontinuities (vertical asymptotes). You can check this by trying to find the value of tan(90°) or tan(π/2) on your calculator (depending if you are in degree or radian mode). We also see that the values for tangent around the unit circle start to repeat at π, which means the fundamental period of tangent is π, as opposed to 2π for sine and cosine.
Tangent clearly looks a lot different than sine and cosine, but the procedure for graphing is similar.
Practice problems for graphing tangent functions: PG 142 #1, 9, 11-14, 19, 21, 27, 32 (for the ones where you need to draw the graph, state the amplitude, period, phase shift, and vertical shift of each cosine function as you set up your graphs)
We don't have an "amplitude" in the same sense we did with sine and cosine, but the coefficient of the tangent function still represents a vertical stretch. We can also still find where the point (0,0) has translated due to a phase shift and/or vertical shift, break the period into 4 sections represented by tick marks, then follow a pattern: zero, nice point (+), vertical asymptote, nice point (-), zero. A negative leading coefficient represents a vertical flip, so the pattern reverses.
Here is the link to the google forms activity for this week. I am going to look more closely at the answers this time and will take points off if your answers are significantly wrong. I have noticed some of you just completely BS'ing these for credit. I am putting a lot of work into this content and it really will benefit you a lot if you dedicate enough time to try to understand what the material. This is more detail than we usually put into the Algebra 2 unit on Trigonometry, and will really be a good launching pad for when you take Pre-calc/Trig in the coming year or so. You can always ask me questions on remind, e-mail, or during my Zoom office hours.