Find a possible formula for the trigonometric function represented by the given table of values. 2 46 8 10 12 3 -1 3 - 1 3 7 3 Preview సా
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
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![**Finding the Trigonometric Function Formula**
In this example, we aim to determine a possible formula for the trigonometric function corresponding to the given table of values.
Here is the table of values provided:
\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\
\hline
y & 3 & -1 & 3 & 7 & 3 & -1 & 3 \\
\hline
\end{array}
\]
To find the possible formula for the trigonometric function, we will analyze the pattern in the given values.
- At \(x = 0\), \(y = 3\)
- At \(x = 2\), \(y = -1\)
- At \(x = 4\), \(y = 3\)
- At \(x = 6\), \(y = 7\)
- At \(x = 8\), \(y = 3\)
- At \(x = 10\), \(y = -1\)
- At \(x = 12\), \(y = 3\)
Given the periodic nature of the data, we will hypothesize a trigonometric function, such as:
\[ y = A \sin(Bx + C) + D \]
or
\[ y = A \cos(Bx + C) + D \]
where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
### Steps to Determine the Constants
1. **Amplitude (A):** The amplitude represents the maximum deviation from the average value. Observing that \(y\) varies from -1 to 7, the amplitude \((A)\) can be estimated as the half of the difference between these extremes.
\[ A = \frac{7 - (-1)}{2} = 4 \]
2. **Midline (D):** The midline represents the average value.
\[ D = \frac{7 + (-1)}{2} = 3 \]
3. **Period (T) and \(B\):** To determine the period, we observe the repeating nature of the values. It seems that the function repeats every 12 units (from \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F416064f9-c57b-46ee-9b50-14b257515654%2F763cb8fd-5274-4bef-8e21-ab6411ef662e%2Fnbpii3m.png&w=3840&q=75)
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