91Ó°ÊÓ

When working with angles, especially in the Cartesian plane, the concept of quadrants becomes very important. The Cartesian plane is divided into four quadrants by the x-axis and y-axis.
Each quadrant is designated by roman numerals and has specific characteristics.

• First Quadrant: Both x and y coordinates are positive.
• Second Quadrant: X coordinates are negative, while y coordinates are positive.
• Third Quadrant: Both x and y coordinates are negative.
• Fourth Quadrant: X coordinates are positive, while y coordinates are negative.

Knowing which quadrant an angle is in helps to determine the sign of trigonometric functions like sine and cosine, which depend on the coordinates of the angle in the plane.
For example, in our exercise, the angle of \(210^{\circ}\) lies in the third quadrant, where both sine and cosine values are negative.

The absolute value in mathematics signifies the distance of a number from zero on the number line, without considering direction.
It is always a non-negative number. When calculating reference angles, using absolute value is crucial because reference angles themselves are always positive.

In our step-by-step solution, we use the formula for the third quadrant: \ |\theta - 180^{\circ}|, to determine the reference angle of \(210^{\circ}\).

• This expression calculates the difference from \(180^{\circ}\), representing how far past the straight line the angle has traveled.
• The absolute value ensures the outcome is positive, making sure the reference angle remains non-negative, conforming to its definition.

Recognizing when and how to use absolute value correctly is essential to ensuring that solutions are both accurate and adhere to mathematical conventions.

Finding angles on a plane involves locating their position or their degree measurement from a specific reference direction, usually the positive x-axis.
A reference angle, however, is slightly different. It is defined as the smallest angle formed between the terminal side of an angle and the x-axis, always measured in positive degrees.

To find a reference angle, a specific formula is applied depending on which quadrant the original angle resides in. For an angle in the third quadrant like \(210^{\circ}\), the formula used is \ |\theta - 180^{\circ}|.

• Here, \(\theta\) is the original angle, \(210^{\circ}\) in our exercise.
• After doing the calculation: \ 210^{\circ} - 180^{\circ} = 30^{\circ}.
• The result, \(30^{\circ}\), is already positive, confirming it's the reference angle.

Learning to calculate reference angles helps simplify trigonometric problems, allowing for easier application of trigonometric functions across any quadrant.