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Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are essential in various areas of mathematics and science. The main trigonometric functions are sine, cosine, and tangent, often abbreviated as sin, cos, and tan. These functions are based on the angles and ratios of sides in a right triangle.

• **Sine (\(\sin\theta\))**: This function represents the ratio of the opposite side to the hypotenuse.
• **Cosine (\(\cos\theta\))**: This function represents the ratio of the adjacent side to the hypotenuse.
• **Tangent (\(\tan\theta\))**: This function represents the ratio of the opposite side to the adjacent side.

These functions are often used in different equations and identities to express relationships between angles and sides. Memorizing these basic definitions is key to solving trigonometry problems. Keep in mind that the Pythagorean identity, one of the most fundamental relationships in trigonometry, states that \(\sin^2 \theta + \cos^2 \theta = 1\). This identity is very helpful when you know one of the values and need to find the other.

Angles on the coordinate plane are divided into four parts, known as quadrants. These quadrants help to determine the sign of trigonometric functions at specific angles.

• **First Quadrant (QI)**: All trigonometric functions are positive.
• **Second Quadrant (QII)**: Sine is positive, while cosine and tangent are negative.
• **Third Quadrant (QIII)**: Tangent is positive, while sine and cosine are negative.
• **Fourth Quadrant (QIV)**: Cosine is positive, while sine and tangent are negative.

In our exercise, \(\theta\) terminates in the fourth quadrant. This implies that the sine of the angle will be negative while the cosine is positive. Knowing the quadrant where the angle terminates is crucial for determining the correct sign of a trigonometric function result. This knowledge allows us to efficiently work with trigonometric values without confusion about their signs.

The relationship between cosine and sine is pivotal in trigonometry. The Pythagorean identity, which states \(\sin^2 \theta + \cos^2 \theta = 1\), shows a fundamental connection between these two functions. This relationship allows you to find the value of one function if you know the other.
In our exercise, we were given \(\cos \theta = \frac{1}{4}\) and needed to find \(\sin \theta\). Here's how the relationship works step-by-step:

• Substitute the known value into the Pythagorean identity.
• Solve the equation to find \(\sin^2 \theta\).
• Since \(\sin^2 \theta\) is the square of sine, take the square root to determine \(\sin \theta\).
• Use the quadrant information to determine the correct sign of \(\sin \theta\).

In this case, since \(\theta\) lies in the fourth quadrant, we choose the negative sign for sine, concluding that \(\sin \theta = -\frac{\sqrt{15}}{4}\). This showcases how understanding the cosine and sine relationship, paired with quadrant information, facilitates solving trigonometric identities with accuracy.