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Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable within their domain. Understanding these identities is crucial for solving trigonometric problems. Some of the most commonly used trigonometric identities include:

• The Pythagorean identities: \(\sin^2 \theta + \cos^2 \theta = 1\), \(1 + \tan^2 \theta = \sec^2 \theta\), and \(\cot^2 \theta + 1 = \csc^2 \theta\).
• Reciprocal identities: \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\).
• Quotient identities: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

These identities are used to simplify and solve trigonometric equations or to relate different trigonometric functions to each other. Mastering these identities will make solving trigonometric problems much easier.

The Pythagorean identity is one of the most important trigonometric identities. It states that for any angle \(\theta\), the sum of the squares of the sine and cosine functions is always equal to 1:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This identity is derived from the Pythagorean theorem and works for all angles \(\theta\). It's incredibly useful for finding unknown values of trigonometric functions. For instance, if you know the value of \(\cos \theta\), you can find \(\sin \theta\) using:
\[ \sin^2 \theta = 1 - \cos^2 \theta\]
In the given problem, we used the identity to find \(\sin \theta\) when we knew \(\cos \theta\). Since we determined that \(\cos \theta = \sqrt{\frac{2}{5}}\), we used the identity to find:
\[ \sin^2 \theta = 1 - \left( \sqrt{\frac{2}{5}}\right)^2 = 1 - \frac{2}{5} = \frac{3}{5}\]
Thus, \(\sin \theta = -\sqrt{\frac{3}{5}}\) because \(\theta\) is in the fourth quadrant where sine is negative.

The coordinate plane in trigonometry is divided into four quadrants, and each quadrant has distinct characteristics for the signs of sine, cosine, and tangent functions.

• First Quadrant: \(0\degree\) to \(90\degree\); \(\sin \theta > 0\), \(\cos \theta > 0\), \(\tan \theta > 0\)
• Second Quadrant: \(90\degree\) to \(180\degree\); \(\sin \theta > 0\), \(\cos \theta < 0\), \(\tan \theta < 0\)
• Third Quadrant: \(180\degree\) to \(270\degree\); \(\sin \theta < 0\), \(\cos \theta < 0\), \(\tan \theta > 0\)
• Fourth Quadrant: \(270\degree\) to \(360\degree\); \(\sin \theta < 0\), \(\cos \theta > 0\), \(\tan \theta < 0\)

In the given problem, since \(\tan \theta = -\frac{\sqrt{6}}{2}\) and \(\cos \theta > 0\), \(\theta\) falls in the fourth quadrant. This means \(\sin \theta\) will be negative, which was crucial for finding the correct value of \(\sin \theta\) in the solution.