91Ó°ÊÓ

The Pythagorean identity is a fundamental concept in trigonometry that provides a relation among the trigonometric functions of an angle within a right triangle. Specifically, the identity states that for any angle \( t \):

\[ \text{For any angle } t, \sin^2(t) + \cos^2(t) = 1. \]
This equation derives from the Pythagorean theorem applied to a unit circle, where the hypotenuse is always 1. When given the value of one trigonometric function, the Pythagorean identity allows us to find the value of another. In our exercise, knowing that \( \cos(t) = -\frac{2}{3} \) and \( t \) is in the second quadrant, we can calculate the positive value of \( \sin(t) \) because in the second quadrant, sine is positive. We use the identity to compute:

\[ \sin^2(t) = 1 - \cos^2(t) \]
\[ \sin^2(t) = 1 - (-\frac{2}{3})^2 \]
\[ \sin(t) = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]
The correct understanding and application of the Pythagorean identity is crucial when trying to determine unknown trigonometric values.

The unit circle is a circle with a radius of 1 centered at the origin \( (0, 0) \) of the coordinate plane. The unit circle is crucial for understanding trigonometric functions across different quadrants. On the unit circle, any point \( (x, y) \) corresponds to \( (\cos(t), \sin(t)) \), where \( t \) is the angle formed by the radius with the positive x-axis.

The angle \( t \) in the exercise slots it into the second quadrant \( (\frac{\pi}{2}