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## Precalculus

### Course: Precalculus > Unit 2

Lesson 8: Sinusoidal models# Interpreting solutions of trigonometric equations

CCSS.Math:

Starting from a context represented by a trigonometric function, interpret equations based on the function. Created by Sal Khan.

## Want to join the conversation?

- I am a bit confused at 3.57, how is the bar at its lowest height when sine is at its max won't it be representing the maximum point the bar can go to since sin 1 is the highest point in the sine function(8 votes)
- No, the solution set 95 = 90 -12 * sin (pi/2) is representing the minimum value, not the maximum.

The equation is B(95)= 90 - 12 sin (pi/2) not 90 +12 *sin (pi/2), therefore it is going to be 90 -12 * 1 = 78.

When it is 90 -12*sin (3pi/2) *(opposite on the unit circle), it is 90 +12 = 112.

If the equation was 90 +12 *sin (pi/2) = 90 + 12 =112 and 90 +12 sin (3pi/2) = 90-12 = 78.

Just think that if the sign before sin/cos function determines what direction it will go to.

If it the sign in front of sin is positive, then when it is sin(pi/2) it will be the maximum. Ex: 12 + sin (pi/2) = 13 or -33 + 30*sin (pi/2) = -3. Also when it is sin (3pi/2) that will be the minimum: Ex: 12 + sin (3pi/2) = 11 or -33 + 30*sin (3pi/2) = -63

If the sign in front of sin in negative, then if it is sin(pi/2) it is the minimum and when it is sin(3pi/3 it is the maximum):

Ex: 41 - 13*sin (pi/2) = 41 -13 = 28

41 - 13*sin (3pi/2) = 41 + 13 = 54

Ex: 79 - 9*sin (pi/2) = 79 - 9 =70

79 - 9*sin (3pi/2) = 79 + 9 = 88

Check this link out to visualize the equation shown above on a graph: https://www.desmos.com/calculator/uqmquflq9m

As you can see, on y = 90 - 12*sin (5t), when x= pi/2, y = 78. On the other graph, y =90 +12*sin (5t), when x=pi/2, it is the maximum value with y =112.

I hope that I was able to explain it well enough to you. Do tell me if you have any questions.(10 votes)

- In my Math career, I have the seen the amplitude refer to an absolute value (distance above or below the midline), but I have also seen it represented with a sign, indicating whether the sinusoidal wave goes up or down first.

So given some equation such as y = 3-4cos(2x + 5*PI), would the amplitude be 4 or -4?

Thanks!(1 vote)- The amplitude is always positive, so the amplitude here is 4. Whether the coefficient on the trig function is positive or negative/whether the wave starts by moving up or down is a different piece of information, separate from the amplitude.(3 votes)

- For the third question, I would have say let pi/2 = 5t, then solve for t, then the solution set is the height of the bar at pi/10 seconds.

But yours works too .. :)(2 votes) - What does the least positive solution mean? For example, what is the least positive solution for 38 = 35 + 9cos(2.4t)? Also, what does it the least positive solution represent in this case?(1 vote)
- A good representation of this problem is thinking of it as a function:

f(x) = 10+9cos(2.4t)

We know that cos(x) is a cycle function, and by it's own f(x)=cos(x) has the highest solution of 1 (where x = 0) and lowest solution of -1 (where x = pi/2).

Therefore, to find the answer to the least positive solution, we just need to find the value where cos(2.4t) = 0. When cos(2.4t) = 0, notice that the only term we have left is 10: the lowest possible solution.

Maybe if this is a temperature function, the least positive solution would be the coldest temperature.(1 vote)

## Video transcript

- [Instructor] Alvaro presses the treadle of a spinning wheel with his foot. It moves a bar up and down
making the wheels spin. So just to be clear what a treadle is this is an old spinning wheel and this little pedal that is a treadle. And as this goes up and down, it's gonna pull on this bar, which is then going to spin this wheel which can then be used to
essentially power the machine. So it says the function B of t models the height in centimeters
of the top of the bar when Alvaro has pressed
the treadle for t seconds. So it's telling us the height of, I can barely see where
the top of the bar is someplace over here. And this isn't exactly what they're probably talking
about in this exercise here. But this is just to
give you a visualization of what a treadle is and what the bar is and then, what the spinning wheel is. Alvaro has pressed those
treadle for t seconds. So they give us B of t right over here and 90 minus 12 times sine of 5t. The first question is, what does the solution set to y is equal to 90 minus 12 times sine of
five times six, represent? Pause this video and see if
you can think about that. All right. So, it looks like right over here, so we have the 90, 90, 12, 12 and we're subtracting 12 sine
of five times t, five times t. So this right over here is t. The solution set right over here tells us what is the height, because
that's what B of t is. So B of t is equal to y. What is the height when t is equal to six? And remember, t is in seconds. So, this is height, height of top of bar, top of bar at six seconds. All right, now we have
more questions here. The next question asks us, what does the solution set to 95 equals 90 minus 12 sign of 5t represent? Pause the video and think about that. All right. So here, they're saying
that B of t is equal to 95. And so, the solution set,
you're really solving for t. So you're really solving
for all of the times when our height is going
to be 95 centimeters. So all times t when height of top of bar, of top of bar at 95 centimeters. And that's going to keep happening over and over and over again
as t goes forward in time. So you're going to have a very large, you're gonna have an infinite
solution set over here. You're gonna have an
infinite number of t's at which your solution at
which the top of the bar is at 95 centimeters. Now we have another question. This one is asking us,
what does the solution set to y is equal to 90 minus 12 sine of pi over two represent? So pause the video and think about that. All right, now this is pretty interesting. We can actually evaluate
what sine of pi over two is. So sign of pi over two
radians or sign of 90 degrees that is going to be equal to one. And so, that's the maximum value that this sign over here can take on. Now, we're going to
subtract 12 times that. So this is taking on a max. Then when you subtract 12 times that this is actually the minimum
value that you can take on. You're gonna have, you can't
get any lower than this. And so, this is going to be
the lowest, the lowest height for the top of the bar. And we're done.