To prove [tex]\(\cos^2(\theta) + \sin^2(\theta) = 1\)[/tex] from the given equation [tex]\(x^2 + y^2 = r^2\)[/tex], follow these steps:
1. Statements: [tex]\(x^2 + y^2 = r^2\)[/tex]
Reasons: Given
2. Statements: [tex]\(\frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2}\)[/tex]
Reasons: Divide each term by [tex]\(r^2\)[/tex]
3. Statements: [tex]\(\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1\)[/tex]
Reasons: Simplified
Continuing from the simplified equation:
- We know from trigonometry that:
[tex]\(x = r \cos(\theta)\)[/tex] and [tex]\(y = r \sin(\theta)\)[/tex]
- Substituting these values into the equation:
[tex]\[
\left(\frac{r \cos(\theta)}{r}\right)^2 + \left(\frac{r \sin(\theta)}{r}\right)^2 = 1
\][/tex]
- This further simplifies to:
[tex]\[
\cos^2(\theta) + \sin^2(\theta) = 1
\][/tex]
Thus, we have proven that [tex]\(\cos^2(\theta) + \sin^2(\theta) = 1\)[/tex].
[tex]\[
\begin{array}{|l|l|}
\hline \text{Statements} & \text{Reasons} \\
\hline 1. x^2 + y^2 = r^2 & \text{1. Given} \\
\hline 2. \frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2} & \text{2. Divide each term by } r^2 \\
\hline 3. \left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1 & \text{3. Simplified} \\
\hline \end{array}
\][/tex]
