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The Pythagorean Identity is a fundamental concept in trigonometry that allows us to connect sine and cosine through their squares. It states:

• \( \sin^2 x + \cos^2 x = 1 \)

This equation is derived from the Pythagorean Theorem and is key in working with trigonometric functions. When you know one trigonometric function, such as sine, you can often find another, like cosine, using this identity.
In the example provided, you start with \( \sin x = -\frac{12}{13} \) in the third quadrant. The Pythagorean Identity helps us compute \( \cos x \) because:

• \( (-\frac{12}{13})^2 + \cos^2 x = 1 \)
• \( \frac{144}{169} + \cos^2 x = 1 \)

Simplifying this gives us \( \cos^2 x = \frac{25}{169} \). Since cosine is negative in the third quadrant, we deduce that \( \cos x = -\frac{5}{13} \). The identity helps ensure that the sum of the squares remains constant, providing consistency across calculations.

The Double Angle Formula is another essential tool in trigonometry that simplifies expressions involving twice an angle. For cosine, this formula is expressed as:

• \( \cos 2x = 2\cos^2 x - 1 \)

In practice, this means that we can transform a trigonometric function of a double angle into an expression involving the square of a single angle's trigonometric function.
In the exercise, after finding \( \cos x = -\frac{5}{13} \), you can use the Double Angle Formula to find \( \cos 2x \). The solution proceeds as follows:

• First, calculate \( 2(-\frac{5}{13})^2 \) to find \( \frac{50}{169} \)
• Substituting this into the formula gives \( \cos 2x = \frac{50}{169} - 1 \)
• Reducing it gives \( \cos 2x = \frac{50}{169} - \frac{169}{169} = -\frac{119}{169} \)

This formula aids in directly relating the angle to its double, allowing simpler computation of trigonometric values for larger angles.

The unit circle is divided into four quadrants, each exhibiting unique properties for the sign of trigonometric functions.

• In Quadrant I, all trigonometric functions are positive.
• In Quadrant II, sine is positive, but cosine and tangent are negative.
• In Quadrant III, both sine and cosine are negative, while tangent is positive.
• In Quadrant IV, cosine is positive, but sine and tangent are negative.

Understanding these quadrants allows us to determine the sign of the sine, cosine, and tangent functions depending on the angle.
For the original problem, knowing \( \sin x = -\frac{12}{13} \) situates us in the third quadrant. In this quadrant:

• \( \sin \) is negative, and \( \cos \) is also negative.

This matches the findings that \( \cos x = -\frac{5}{13} \). Proper interpretation of the quadrants is vital for predicting the behavior and relationship of these functions, no matter the size of the angle.