Can you please help me solve cosine and sine cicular functions with translations on x and dialations on x in the domain -pi and pi

Certainly! Let's go through the process of solving cosine and sine circular functions that involve translations and dilations on the x-axis, within the domain \([-π, π]\).

### 1. **Basic Form of Sine and Cosine Functions**
The general form of the sine and cosine functions with translations and dilations is:

\[
y = A \cdot \sin(B(x - C)) + D
\]
\[
y = A \cdot \cos(B(x - C)) + D
\]

- **\(A\)** is the amplitude (vertical stretch/compression).
- **\(B\)** affects the period (horizontal stretch/compression).
- **\(C\)** is the horizontal translation (shift).
- **\(D\)** is the vertical translation (shift).

### 2. **Horizontal Dilation**
Horizontal dilation is controlled by the value of \(B\). The period of the function changes with \(B\):

\[
\text{Period} = \frac{2\pi}{|B|}
\]

For example, if \(B = 2\), the period of the function is \(\frac{2\pi}{2} = \pi\).

### 3. **Horizontal Translation**
The horizontal translation is determined by \(C\). If \(C > 0\), the graph shifts to the right by \(C\) units. If \(C < 0\), it shifts to the left by \(|C|\) units.

### 4. **Solving within the Domain \([-π, π]\)**
When solving for \(x\) within the domain \([-π, π]\), you'll need to:
1. **Determine the period**: Based on \(B\), calculate how the function's period fits within \([-π, π]\).
2. **Account for translations**: Apply the translation \(C\) to determine the shifted domain.
3. **Solve for \(x\)**: For specific function values, solve the equation for \(x\) by isolating it on one side.

### 5. **Example Problem**

Let's consider the function:

\[
y = \cos\left(2\left(x - \frac{\pi}{4}\right)\right)
\]

#### **Step 1: Period and Dilation**
The period is determined by \(B = 2\):

\[
\text{Period} = \frac{2\pi}{2} = \pi
\]

#### **Step 2: Horizontal Translation**
The function is translated to the right by \(\frac{\pi}{4}\) units.

#### **Step 3: Solve for \(x\)**
Let's say we want to find where \(y = 0\) within \([-π, π]\):

\[
\cos\left(2\left(x - \frac{\pi}{4}\right)\right) = 0
\]

\[
2\left(x - \frac{\pi}{4}\right) = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}
\]

Divide by 2:

\[
x - \frac{\pi}{4} = \frac{\pi}{4} + \frac{n\pi}{2}
\]

Add \(\frac{\pi}{4}\) to both sides:

\[
x = \frac{\pi}{2} + \frac{n\pi}{2}
\]

So the general solutions for \(x\) within the domain \([-π, π]\) are:

\[
x = \frac{\pi}{2} + \frac{n\pi}{2}
\]

Check for specific values within the domain \([-π, π]\):

- \(n = -3\), \(x = -\frac{3\pi}{2} + \frac{\pi}{2} = -\pi\) (not in the domain)
- \(n = -2\), \(x = -\pi + \frac{\pi}{2} = -\frac{\pi}{2}\) (valid)
- \(n = -1\), \(x = -\frac{\pi}{2} + \frac{\pi}{2} = 0\) (valid)
- \(n = 0\), \(x = \frac{\pi}{2}\) (valid)
- \(n = 1\), \(x = \frac{\pi}{2} + \frac{\pi}{2} = \pi\) (valid)

Thus, \(x\) could be \(-\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi\) within \([-π, π]\).

### Questions for Further Exploration
1. How does the period of a sine function change if \(B\) is a fraction?
2. What happens to the graph of the function if \(C = 0\)?
3. How would you solve the sine function \(y = \sin(3(x + \frac{\pi}{6}))\) within \([-π, π]\)?
4. Can you explain the effect of vertical translations on the sine and cosine functions?
5. What are the specific solutions for \(x\) if \(y = \cos(x)\) and \(B = \frac{1}{2}\)?

### Tip
Remember that horizontal dilation affects the speed of the oscillation: a larger \(B\) makes the function oscillate more quickly, reducing the period.

Learn how to solve cosine and sine circular functions involving translations and dilations on the x-axis within the domain [-π, π]. Understand the effects of amplitude, period, horizontal translation, and vertical translation on these functions. Suitable for high school students.

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