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## Trigonometry

### Course: Trigonometry>Unit 4

Lesson 3: Sinusoidal models

# Trig word problem: solving for temperature

Sal solves a word problem about the annual change in temperature by solving a sinusoidal equation. Created by Sal Khan.

## Want to join the conversation?

• At , Sal says"In the last video,
we were able to model ……". Anyone can teach me How to find that video?
• I was doing the exercise after this video and I need some help:

Antonio's toy boat is bobbing in the water under a dock. The vertical distance H (in cm) between the dock and the top of the boat's mast t seconds after its first peak is modeled by the following function. Here, t is entered in radians.

`H(t)* = 5cos(2π/3 * t ) - 35.5`*

How long does it take the toy boat to bob down from its peak to a height of −35 cm?*

Here's the answer to solve the equation:

`H(t) = 5cos(2π​​/3 * t) - 35.5` has a period of `2π​​/2π​​/3 = 3 seconds`.
We want to find the first solution to the equation `H(t) = -35` within the period `0 < t < 3`.

`5cos(2π/3 * t) - 35.5 = H(t)5cos(2π/3 * t) - 35.5 = -355cos(2π/3 * t) = 0.5cos(2π/3 * t) = 0.5/5`
Rounding the several decimal places (in order to be precise), we get *cos^-1(0.5/5)* ≈ 1.4706

`2π/3 * t = 1.4706+2πnt ≈ 0.7+3n`
Now, let's use the identity and cos(θ) = cos(2π−θ)* to find the second expression for all possible values of *t.
`2π/3 * t = (2π − 1.4706) + 2πnt = 3/2π * (2π - 1.4706 + 2πn)t ≈ 2.7+3n`

Therefore, it takes about 0.7 seconds for the boat to bob down to -35cm.
*********************************************************************************************************
My questions are:

1. Where did the n come from in the 2π*?
2. Why is *2π* added in after getting *1.4706
?

I hope my question get responded as soon as possible. I'm really struggling to understand this.
Thanks.
• The n is a variable without an answer in this question. 2pi is the same as 2*180 or 360 because pi=180 when using degrees and radians. The sin, cos or tan of any angle will be the same when you add 360 degrees or 2pi. So they added the n so that you know that it could be 1.4706 or 1.4706 times 2pi or 1.4706 times 2*2pi, etc. All of those values will be the same. (It would also be the same if you added 360 or 2*360 or 3*360, etc)
Does that help?
• So when he does the cos^-1(-0.2) he's getting 1.77~. Whenever I do it though, I keep getting 101.~. I've checked it on 3 different calculators and it all says the same thing. How is he getting that?
• The formula is specific to radian measure, so that's how Sal is doing his calculations. 1.77 radians is the same as 101 degrees.
• Towards the end of the video, I input the same equation you display on screen, but i keep getting 2586.339714 as my answer. I looked below and saw someone in a similar situation, so I tried inputting 2 * pi instead. I ended up with the same answer. This also happens in the practice problems that I try and I'm not sure what I'm doing wrong. The steps I took are the same as in the video since I was following along.
• I was having the same issue! Instead of doing Ans*365/2pi, you have to do Ans*365/(2pi). for some reason the calculators I've used do 4.511*365/2 then multiplies it by pi which gets you 2586 days rather than 262. I'm not sure how Sal's calculator does it correctly but that's the only way I've found to make it work. 7 years late but hopefully this can help somebody else!
• This video says the last video.
Why can't I see where the last video is?
• You start the video by saying "In the last video we..." It would be helpful if there were a link to that last video. I couldn't find it.
• I didn't understand the last step where Sal said that the value of 2pi/365*d=4.511.
• (1) We are trying to find where cos(2pi/365*d) = -0.2, but there are two values of (2pi/365*d) in the year that satisfy this.
(2) The two values of x that satisfy cos(x) = -0.2 are x = 1.772 and x = 4.511.
(3) By substituting 2pi/365*d for x, we know that either 2pi/365*d=1.772 or 2pi/365*d=4.511.
(4) From Sal's unit circle work, we know we want the one in quadrant III.
(5) 4.511 falls in quadrant III, so the equation we want is 2pi/365*d=4.511.

Hopefully this helps.
• In some of the Trig lessons I see sin(θ)=sin(π−θ) and in others I see sin(θ)=sin(-π−θ). How do I know which property to use?
• Just use whatever property need for situation.

A couple of important identities that you may not have learned yet are:

-sin(theta)=sin(-theta)
-tan(theta) = tan(-theta)
cos(theta) = cos(-theta)

The functions tan and sin are considered odd because -f(x) = f(-x). And Cos is even since f(x)=f(-x). So in this case sin(pi-theta)= - sin(theta-pi) since sine is odd. So base on this we can tell the sine function shifted pi to the right as each value needs have pi added to it retain the same value as sin theta. After this graph flip across the x-axis due to minus sign in front sin(theta-pi). So you can visually see that sin theta = sin(pi-theta).

Now on to the second identity all you just doing shifting the function pi to the left, and since sin has period 2pi what this means is shifting function by the same amount.

So you will need to use the identities depending on the context as there is no difference between them.
• I am have some trouble with these practice problems... the video is nothing like the practice problems after it. At there is usually at 2pi * n at the end with the practice problems but not this video. so like t = 365/2pi * (0.9273 + 2pi* n) Anyone now how to help me solve this?

At , how does Sal calculate the fraction 1.5/1.7 in his head so fast? I need to develop number sense like that!