91Ó°ÊÓ

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane.

Every point on the unit circle corresponds to an angle, measured from the positive x-axis. For example, an angle of 0 degrees points directly to the right, along the positive x-axis. Meanwhile, a 90-degree angle points straight up, along the positive y-axis.

On the unit circle, the coordinates of each point are given by \(x, y\), where:

• x is the cosine of the angle, and
• y is the sine of the angle.

In this system, every angle also has a corresponding angle in each of the four quadrants. When the angle increases, the point moves clockwise around the circle.

When given an angle like 225°, you can locate it on the unit circle. Since it's more than 180° but less than 270°, it lies in the third quadrant, where both sine and cosine values are negative.

A reference angle is a useful tool in trigonometry for finding the sines and cosines of larger angles.

Essentially, a reference angle is the smallest angle that the given angle makes with the x-axis. It's always an acute angle, which means it's between 0° and 90°.

To find a reference angle, you subtract the closest x-axis from the given angle. For example, with an angle of 225°, the nearest x-axis lies at 180°. So the reference angle is 225° - 180°, which equals 45°.

Reference angles allow us to use the known trigonometric values for these smaller, more manageable angles, applying them to angles found in different quadrants. When the reference angle is known, we can determine the trigonometric functions' signs based on which quadrant the original angle lies in.

The cosine function, denoted as cos(θ), is one of the six fundamental trigonometric functions. It relates an angle in a right-angled triangle to the ratio of the adjacent side over the hypotenuse.

In the context of the unit circle, the cosine function gives you the x-coordinate of a point corresponding to the angle θ.

For instance, the cosine of 45° is \(\frac{\sqrt{2}}{2}\), and this value applies to all angles that have a reference angle of 45°.

However, remember that the cosine value changes sign depending on the quadrant of the angle. For angles like 225° that reside in the third quadrant, the cosine is negative.

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive; cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative; cosine is positive.

Thus, for an angle of 225°, the cosine is -\(\frac{\sqrt{2}}{2}\), due to its position in the third quadrant.