91Ó°ÊÓ

In trigonometry, the coordinate plane is divided into four areas known as quadrants. These quadrants help in determining the sign (positive or negative) of trigonometric functions for different angles.

The quadrants are numbered from I to IV in an anti-clockwise direction:

• **First Quadrant (I)**: Both sine and cosine are positive.
• **Second Quadrant (II)**: Sine is positive, but cosine is negative.
• **Third Quadrant (III)**: Both sine and cosine are negative.
• **Fourth Quadrant (IV)**: Sine is negative, but cosine is positive.

The angle \(\frac{7\pi}{4}\) lies in the **fourth quadrant**. To verify, we see it's between \(\pi\) and \(2\pi\), confirming it's in Quadrant IV.

The sine function, often referred to as sin(θ), is one of the fundamental trigonometric functions. It measures the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.

To understand the sine function for angles in different quadrants, remember:

• **First Quadrant**: Positive
• **Second Quadrant**: Positive
• **Third Quadrant**: Negative
• **Fourth Quadrant**: Negative

Since \(\frac{7\pi}{4}\) is in the fourth quadrant, we know that sin(θ) will be negative. This aligns perfectly with the rule. But, we first need its reference angle.

Reference angles help us find the exact trigonometric values when the angle is not in the first quadrant.

Finding the exact values of trigonometric functions involves reference angles. Here’s how we determine them using our example:

For \(\frac{7\pi}{4}\) in the fourth quadrant, the reference angle is calculated as \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\). This reference angle is within the first quadrant and is a standard angle from the unit circle.

The sine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\)

Since our original angle lies in the fourth quadrant, where sine is negative, we assign a negative sign to the value. Therefore, \(\sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2}\)

Using reference angles, mastering these conversions and function values will significantly boost your understanding of trigonometry.