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## Trigonometry

### Course: Trigonometry > Unit 2

Lesson 10: Sinusoidal models- Interpreting trigonometric graphs in context
- Interpreting trigonometric graphs in context
- Trig word problem: modeling daily temperature
- Trig word problem: modeling annual temperature
- Modeling with sinusoidal functions
- Trig word problem: length of day (phase shift)
- Modeling with sinusoidal functions: phase shift

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# Interpreting trigonometric graphs in context

When a trigonometric function models a real-world relationship, we can assign meaning to its midline, amplitude and period. Created by Sal Khan.

## Want to join the conversation?

- I thought the practice from the last chapter said that midline is 1/4 of a period, so 4 x 15 right? AHhhhh i keep confusing myself(4 votes)
- No, 4x10, because the midline is x=15 and max is x=25, min is x = 5. 25-15 = 10, and 15-5=10 as well.

Generally its 4 x difference between x of midline and x of the immediate next min or max.(7 votes)

- I didn't understand why the mid-line represents the center of rotation?(3 votes)
- The midline represents the average of the max_Yvalue, min_Yvalue(7 votes)

- I never knew that Ferris wheels went that fast...(2 votes)

- Is the period always the same?(1 vote)
- Not necessarily. Functions can have different periods(2 votes)

- can you express all sinusoid functions with both cosine and sine functions or every sinusoid function can only be expressed with either sine or cosine?(1 vote)
- Any function that can be expressed in terms of a sine function can also be expressed as a cosine function (and vice versa).(1 vote)

- I'm confused, if in the sin(x) function the independent variable corresponds to an angle in radians, why is the independent variable a duration in this video?(1 vote)

## Video transcript

- [Instructor] We're told Alexa
is riding on a Ferris wheel. Her height above the ground in meters is modeled by H of t where
t is the time in seconds. And we can see that right over here. Now what I want to focus on this video is some features of this graph. And the features we're gonna focus on, actually the first of them,
is going to be the midline. So pause this video and
see if you can figure out the midline of this graph, or
the midline of this function. And then we're going to think about what it actually represents. Well, Alexia starts off at
five meters above the ground and then she goes higher,
and higher, and higher, and gets as high as 25 meters, and then goes back as low as
five meters above the ground, and as high as 25 meters. And what we can view the midline as is the midpoint between these extremes, or the average of these extremes. Well, the extremes are
she goes as low as five and as high as 25. So what's the average of five and 25? Well, that would be 15. So the midline would
look something like this. And I'm actually gonna
keep going off the graph. And the reason is is
to help us think about what does that midline even represent. And one way to think about it is it represents the center of
our rotation in this situation, or how high above the ground is the center of our Ferris wheel? And to help us visualize that,
let me draw a Ferris wheel. So I'm going to draw a circle
with this as the center. And so the Ferris wheel would
look something like this. And it has some type of
maybe support structure. So the Ferris wheel might
look something like that. And this height above the ground, that is 15 meters, that is what
the midline is representing. Now the next feature I want
to explore is the amplitude. Pause this video and think
about what is the amplitude of this oscillating
function right over here and then we'll think about
what does that represent in the real world, or where does it come
from in the real world? Well the amplitude is
the maximum difference or the maximum magnitude
away from that midline. And you can see it right over here. Actually right when Alexa starts we have starting 10
meters below the midline, 10 meters below the center, and this is when Alexa is right over here. She is 10 meters below the midline. And then after, it looks like 10 seconds, she is right at the midline. So that means that she is right over here. Maybe the Ferris wheel is going this way, at least in my imagination
it's going clockwise. And then after another 10
seconds she is at 25 meters. So she is right over there. And you could see that. She is right over there, I
drew that circle intentionally of that size. And so we see the
amplitude in full effect. 10 meters below to begin the
midline and 10 meters above. And so it's the maximum displacement or the maximum change from that midline. And so over here it really
represents the radius of our Ferris wheel, 10 meters. And then from this part she
starts going back down again. And then over here she's
back to where she started. Now the last feature I want to explore is the notion of a period. What is the period of
this periodic function? Pause this video and think about that. Well the period is how
much time does it take to complete one cycle. So here she's starting at the bottom. And let's see, after 10
seconds, not at the bottom yet. After 20 seconds, not at the bottom yet. After 30 seconds, not at the bottom yet. And then here she is, after 40 seconds, she's back at the bottom
and about to head up again. And so this time right
over here, that 40 seconds, that is the period. And if you think about
what's going on over here, she starts over here five
meters above the ground. After 10 seconds she is right over here, and that corresponds to
this point right over here. After 10 more seconds
she's right over there. That corresponds to that point. After 10 more seconds she's over here. That corresponds to that. And after 10 more seconds,
or total of 40 seconds, she's back to where she started. So the period in this example shows how long does it take
to complete one full rotation. Now we have to be careful sometimes when we're trying to
visually inspect the period because sometimes it might be tempting to say, "Start right over here, "and say okay, we're 15
meters above the ground. "All right, let's see, we're going down. "Now we're going up again. "And look, we're 15
meters above the ground. "Maybe this 20 seconds is a period." But when you look at it over here it's clear that that is not the case. This point represents this point at being 15 meters above the ground, going down, that's
getting us to this point, and then after another 10
seconds we get back over here. Notice, all this is
measuring is half of a cycle going halfway around. In order to go all the way around not only do we have to get
to the same exact height, but we have to be moving
in the same direction. We're at 15 meters and going down. Here we're at 15 meters and going up. So we have to keep going anther 20 seconds in order to be 15 meters
in the air, and going down.