91Ó°ÊÓ

The unit circle is divided into four quadrants. Each quadrant has specific characteristics for the sine, cosine, and tangent values of angles. Understanding which quadrant an angle is in helps you determine the sign of these trigonometric functions.

The quadrants are organized as follows:

• First Quadrant (0 to \(\frac{\pi}{2}\)): Both sine and cosine values are positive.
• Second Quadrant (\(\frac{\pi}{2}\) to \(\frac{\pi}{2}\)): Sine is positive and cosine is negative.
• Third Quadrant (\(\pi\) to \(\frac{3\pi}{2}\)): Both sine and cosine values are negative.
• Fourth Quadrant (\(\frac{3\pi}{2}\) to \(\frac{2\pi}{2}\)): Sine is negative while cosine is positive.

In this exercise, \(\frac{3\pi}{4}\) lies in the second quadrant. Thus, the cosine value for this angle will be negative.

A reference angle is formed by the terminal side of the angle and the horizontal axis (the closest x-axis). Reference angles are useful because they allow us to use known values from the first quadrant to find the values in other quadrants.

To find the reference angle when given an angle in the second quadrant, subtract the angle from \(\pi\) (or 180°). For example, the reference angle for \(\frac{3\pi}{4}\) is calculated as follows: \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\). Thus, the reference angle of \(\frac{3\pi}{4}\) is \(\frac{\pi}{4}\).

The cosine function measures the horizontal distance from the origin to the point on the unit circle. The cosine of an angle is related to its reference angle, but you must also consider the sign based on the quadrant.

In the second quadrant, the cosine value is negative. Using the reference angle \(\frac{\pi}{4}\), we know that \(\cos\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}} \text{or} \frac{\sqrt{2}}{2}\). Since the angle \(\frac{3\pi}{4}\) is in the second quadrant, its cosine value will be: \(\cos\left( \frac{3\pi}{4} \right) = - \frac{1}{\sqrt{2}} = - \frac{\sqrt{2}}{2}.\)

Remember that knowing the reference angle and the quadrant helps make finding exact values for trigonometric functions much easier.