Question 1

Did the modeling w/ sinusoidal functions (phase shift) unit; how to know when to do sin or cos since either can happen w/ a phase shift? Instructions don't specify which/my answer satisfies requirements, but it isn't counted as correct.

Accepted Answer

As we know, when looking at a unit circle, cosine's value will reach 1 when x=0 radians, whereas sine's value will reach 1 when x=pi/2 radians. In a scenrio where the graph reaches its maximum point when x=0, you know that it will be a graph of cosine. However, if you have a scenario where you reach the midline (midpoint) at a x-value of 0, you know you have a graph of sine. Essentially, you want to look at where you reach your maximum or midline when analyzing what type of a graph to use. Now, if you are given a problem with a phase shift, you want to look at the maximum and minium and shift it with relevance to what format (sine or cousin) your graph is in. I would suggest looking at one of the videos that Sal posted. He really thoroughly explains how to know what format to use.

Question 2

I really do not understand the substitution in phase shifting a function. I've tried it in every way I can imagine solving it... for example: −1.5cos(2π/248(22+11))+5.9 , solved in several different ways, always evaluated to approx. 4.4002.... I got those questions wrong every single time. what am I doing wrong here? what am I missing??

Accepted Answer

could you show me your work? what are the steps you take to get this answer? Also, what is the problem? This is how I would solve it:

1)(33π/248) ≈ 0.836069...

2) cos(0.836069) ≈ 0.6703848... *MAKE SURE TO USE RADIAN MODE FOR THIS STEP*. The khan academy calculator(and most others) have a deg and rad mode for trig functions.

3) -1.5*0.670384... ≈ -1.0055..

4) -1.0055...+5.9 = 4.894422...

Hope this helped! Next time show your work as well :)

Question 3

I am struggling to understand one of the problems I encountered where I was asked WHEN the function would hit it's maximum.

There was a function given,

```
h(t) = 5 - 2 * sin(2pi (t + 1) / 7)
```

I understand that the function is negative so the it's a sine reflection.

I think I also understand that the function ignoring the +1 would reach it's maximum at t=5.25 (i.e. 3/4 * 7)

what I don't understand is why when considering the +1 the max takes place at t=6.25 instead of t=4.25?

Shouldn't a +1 SHIFT the whole function to the LEFT - causing the MAX to be encountered sooner by 1?

Could someone please help me with this fundamental gap in my understanding?

Thank you.

Accepted Answer

unless you are reading the problem wrong(which happens to the best of us), then you are right and the answer is wrong. It should be 4.25. Maybe it's -1 and not +1?

Question 4

On one of the practice quizzes the answer in part only was "-357cos(...". I came up with the same answer except I had as positive instead of a negative, i.e. "357cos(...". I tried them both on Desmos and the -357 did not appear correct to me. Did I miss something? I admit some of these problems have been a challenge.

Accepted Answer

the negative flips the graph across the x-axis. This could mean that in your problem, the starting value needed to be the minimum point(this might not be the case since you did not provide the question)

Question 5

So I'm just kind of confused about this. On one of the questions in the modeling with sinusoidal functions:

In January, the average temperature t hours after midnight in Mumbai, India, is given by:

T(t) = 24.5 - 5.5sin(2pi(t + 1)/24)

What is the coldest time of day in Mumbai?

The hints give me this:

sin u is largest when u is pi/2 plus a multiple of 2pi. So the coldest time of day is when
2pi(t + 1)/24 = pi/2 + 2pi*n

For n an integer

We can solve the equation for t:

t + 1 = 6 + 24n
t = 5 + 24n

My question is, where did they get the 6? And also is there a reason why the sin u formula is used specifically?

Accepted Answer

Starting from this equation:
2pi(t + 1)/24 = pi/2 + 2pi*n

we can divide everything by pi:
2(t+1)/24= 1/2 + 2n

and multiply everything by 12 to clear denominators:
t+1 = 6 + 24n

The 6 is the result of (1/2)·12. Notice that we simplified the fraction on the left-hand side as well.

The formula is sinusoidal because the temperature is going up and down the same amount at regular intervals. It uses sine instead of cosine for no particular reason; any equation with sine can be rewritten with cosine instead, and vice-versa, using sin(x)=cos(π/2-x).

Question 6

Is a sine function a phase shift of a cosine function and vice versa?

Accepted Answer

Yes. sin(x+π/2)=cos(x).

Course: Algebra 2 > Unit 11

Trigonometry: FAQ

What is the unit circle and why is it important in trigonometry?

What are radians and why do we use them in trigonometry?

What is the Pythagorean identity and why is it important?

What do amplitude, midline, and period mean when we're talking about sinusoidal graphs?

Why do we want to know how to transform sinusoidal graphs?

Where are trigonometric functions used in the real world?

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