Let's fill out the table with the given values.
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\theta & \sin \theta & (\sin \theta)^2 & \cos \theta & (\cos \theta)^2 & (\sin \theta)^2 + (\cos \theta)^2 \\
\hline
\frac{\pi}{4} & 0.7071067811865475 & 0.4999999999999999 & 0.7071067811865476 & 0.5000000000000001 & 1.0 \\
\hline
\frac{2 \pi}{3} & 0.8660254037844387 & 0.7500000000000001 & -0.4999999999999998 & 0.24999999999999978 & 0.9999999999999999 \\
\hline
\frac{7 \pi}{6} & -0.4999999999999997 & 0.24999999999999972 & -0.8660254037844388 & 0.7500000000000003 & 1.0 \\
\hline
\frac{5 \pi}{3} & -0.8660254037844386 & 0.7499999999999999 & 0.5000000000000001 & 0.2500000000000001 & 1.0 \\
\hline
2 \pi & -2.4492935982947064 \times 10^{-16} & 5.99903913064743 \times 10^{-32} & 1.0 & 1.0 & 1.0 \\
\hline
\end{array}
\][/tex]
Now let's analyze the patterns.
### Do You See a Pattern?
Yes, there is a noticeable pattern in the values of [tex]\((\sin \theta)^2 + (\cos \theta)^2\)[/tex]. For each angle [tex]\(\theta\)[/tex], the value of [tex]\((\sin \theta)^2 + (\cos \theta)^2\)[/tex] equates to 1, or practically very close to 1 considering numerical precision in computations.
### Do You Think This Will Always Be True? Why or Why Not?
Yes, this pattern will always be true. This is because of a fundamental trigonometric identity known as the Pythagorean identity. According to this identity, for any angle [tex]\(\theta\)[/tex]:
[tex]\[ (\sin \theta)^2 + (\cos \theta)^2 = 1 \][/tex]
This identity comes from the definition of sine and cosine in the unit circle. In the unit circle, any point [tex]\((x, y)\)[/tex] on the circle satisfies [tex]\( x^2 + y^2 = 1 \)[/tex], where [tex]\(x\)[/tex] is [tex]\(\cos\theta\)[/tex] and [tex]\(y\)[/tex] is [tex]\(\sin\theta\)[/tex]. Hence, you always get:
[tex]\[ (\cos \theta)^2 + (\sin \theta)^2 = 1 \][/tex]
Thus, no matter the angle [tex]\(\theta\)[/tex], [tex]\((\sin \theta)^2 + (\cos \theta)^2\)[/tex] will always be 1.
