# What Is the Domain and Range of y=cos(3(x - 45°)) +2?

• MHB
• mathuravasant
In summary, the domain for one cycle of y=cos(3(x - 45°)) +2 is 0 ≤ x ≤ 360 and the range is 1 ≤ y ≤ 3. The function can be graphed to better understand the values of the domain and range. It is also helpful to have a clear understanding of the definitions of domain, range, frequency, wave number, horizontal shift, and vertical shift.
mathuravasant
State the domain and range for one cycle of y=cos(3(x - 45°)) +2 Show your work.

Beer induced request follows.
mathuravasant said:
State the domain and range for one cycle of y=cos(3(x - 45°)) +2 Show your work.
Please show us what you have tried and exactly where you are stuck.

like how do you find domain and range off from that given equation I just don't know what to do

mathuravasant said:
like how do you find domain and range off from that given equation I just don't know what to do
You are having a great deal of trouble with these. I suspect the biggest cause isn't in the problems but in the definitions. I would suggest having a 3 x 5 card (or some other modern equivalent) stating what the domain, range, frequency, wave number, horizontal shift (or phase), and vertical shift are for each.

$$\displaystyle y = cos( 3(x - 45) ) + 2$$.

What is the domain? It's the interval on the x-axis that the function is defined on. So one cycle is 360 degrees. Thus
$$\displaystyle 0 \leq 3(x - 45) \leq 360$$
So what are the possible values for x? $$\displaystyle 0 \leq 3(x - 45)$$ to $$\displaystyle 3(x - 45) \leq 360$$

What is the range? It's the interval on the y-axis that the function takes on over the domain. So for the sake of argument let's say that the domain is $$\displaystyle -45 \leq x \leq 90$$. (Mind you, it isn't.) Then what is the range of cosine? It's best to graph this one and take a look since cosine "waves" so graph $$\displaystyle y = cos( 3(x - 45) ) + 2$$ and find the biggest change in y value for $$\displaystyle -45 \leq x \leq 90$$.

Let us know how it goes.

-Dan

## 1. What is a sinusoidal function?

A sinusoidal function is a mathematical function that describes a periodic oscillation, such as a wave or vibration. It is represented by the equation f(x) = A sin (Bx + C) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

## 2. What is the difference between a sinusoidal function and a cosine function?

A cosine function is a type of sinusoidal function that has a phase shift of 0 degrees, meaning it starts at the maximum value. A general sinusoidal function can have any phase shift, which will affect the starting point of the function. Additionally, the graphs of these functions differ slightly in shape, with the cosine function being slightly shifted to the left.

## 3. How do I find the period of a sinusoidal function?

The period of a sinusoidal function is the length of one complete cycle of the function. It can be calculated by dividing 2π by the coefficient of x in the equation, or by finding the distance between two consecutive maximum or minimum points on the graph.

## 4. What is the relationship between sinusoidal functions and trigonometric ratios?

Sinusoidal functions are closely related to trigonometric ratios, specifically the sine and cosine functions. The sine function represents the vertical displacement of a point on a unit circle, while the cosine function represents the horizontal displacement. These functions can be used to model sinusoidal functions and vice versa.

## 5. How are sinusoidal functions used in real life?

Sinusoidal functions have many real-life applications, such as modeling sound waves, ocean tides, and electromagnetic waves. They are also used in fields such as engineering, physics, and music to analyze and predict the behavior of periodic phenomena.

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