91Ó°ÊÓ

The Pythagorean identity is a fundamental relation in trigonometry that connects the sine and cosine functions. It states that for any angle \( t \), the following equation holds:\[\sin^2 t + \cos^2 t = 1\]This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. Understanding this identity allows for the calculation of one trigonometric function if the other is known. For example, in the exercise, knowing \( \sin t = -\frac{2}{3} \) lets us find \( \cos t \) by rearranging the identity:

• First, calculate \( \sin^2 t \) as \( \left(-\frac{2}{3}\right)^2 = \frac{4}{9} \).
• Then substitute into the identity: \( \frac{4}{9} + \cos^2 t = 1 \).
• Solving for \( \cos^2 t \), we get \( \cos^2 t = \frac{5}{9} \).

Finally, by taking the square root, we find that \( \cos t = \pm\frac{\sqrt{5}}{3} \). Since the function signs depend on the quadrant, we have to consider them next.

In trigonometry, the coordinate plane is divided into four quadrants, each affecting the signs of trigonometric functions. These quadrants are:

• 1st Quadrant: Sine, cosine, and tangent are all positive.
• 2nd Quadrant: Sine is positive, cosine and tangent are negative.
• 3rd Quadrant: Sine and cosine are negative, tangent is positive.
• 4th Quadrant: Cosine is positive, sine and tangent are negative.

In this exercise, the angle \( t \) is located in the third quadrant. Knowing the quadrant helps determine the correct signs for \( \sin t \) and \( \cos t \). Both are negative in the third quadrant, which is crucial for solving problems like this one, ensuring the correct sign is chosen for \( \cos t \). Therefore, even though \( \cos t = \pm\frac{\sqrt{5}}{3} \) was found, the negative root \( -\frac{\sqrt{5}}{3} \) is selected due to its location in the third quadrant.

The sign of trigonometric functions depends greatly on the quadrant in which the angle lies. Each quadrant imposes a rule on whether the sine, cosine, and tangent of angles are positive or negative. Let's review some key points:

• Sine: Positive in the 1st and 2nd quadrants, negative in the 3rd and 4th quadrants.
• Cosine: Positive in the 1st and 4th quadrants, negative in the 2nd and 3rd quadrants.
• Tangent: Positive in the 1st and 3rd quadrants, negative in the 2nd and 4th quadrants.

In the context of the exercise, this understanding allows us to correctly apply the Pythagorean identity and select the right sign for the trigonometric functions. Specifically, since \( t \) is in the third quadrant, we recognize:

• \( \sin t \) is negative, as initially given.
• \( \cos t \) must also be negative, confirmed by choosing the correct root when applying the Pythagorean identity.

This not only helps in verifying solutions but also ensures a solid grasp of trigonometry's foundational concepts.