Question

Find the reference angle of each angle. $$ -30^{\circ} $$

Short answer

30°

Step-by-step solution

Understand the Definition of a Reference Angle

A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always positive and lies between 0° and 90°.

Identify the Quadrant of the Angle

-30° is in the fourth quadrant as it is measured clockwise from the positive x-axis.

Calculate the Reference Angle

For an angle in the fourth quadrant, the reference angle is found by taking the absolute value of the given angle. Hence, the reference angle for -30° is \(|-30^{\text{°}}| = 30^{\text{°}}\text{.} \)

Key concepts

Reference Angle Definition

In trigonometry, a reference angle is essential for simplifying the calculations with angles. A reference angle is the smallest angle formed between the terminal side of the given angle and the x-axis. It is always a positive value and lies between 0° and 90°. This makes it easier to work with since reference angles are always acute angles. Remember, the terminal side is the ray where the angle ends. For example, if we consider an angle of 120°, its reference angle would be 60°, since 180° - 120° = 60°.

Quadrant Identification

The position of an angle in the coordinate plane is significant because it helps in determining its reference angle. Each angle lies in one of four quadrants:

- Quadrant I: Angles from 0° to 90°.
- Quadrant II: Angles from 90° to 180°.
- Quadrant III: Angles from 180° to 270°.
- Quadrant IV: Angles from 270° to 360°.

A negative angle indicates a clockwise direction from the positive x-axis. For instance, -30° is in the fourth quadrant. Identifying the correct quadrant helps in calculating the reference angle accurately.

Angle Calculation

To find the reference angle, the method depends on the quadrant:

- **First Quadrant**: The reference angle is the angle itself.
- **Second Quadrant**: Subtract the angle from 180°.
- **Third Quadrant**: Subtract the angle from 180°.
- **Fourth Quadrant**: Take the absolute value of the angle.

Let's apply this to our example. For an angle of -30°, we know it is in the fourth quadrant. Hence, the reference angle will be the absolute value of -30°, which is 30°.
This rule of thumb works for both positive and negative angles.

Absolute Value

The absolute value is a crucial concept in mathematics and helps significantly in angle calculations. The absolute value of a number is its distance from zero on the number line, disregarding the sign. Therefore, \(|\text{-30°}| = 30°\). When dealing with angles, the absolute value ensures that we always work with positive measures when calculating reference angles. For example, irrespective of whether an angle is -45° or +45°, its absolute value will be 45°, simplifying our computations. Always remember: the reference angle is between 0° and 90° and should always be positive.