Question

Find the reference angle of each angle. $$ 210^{\circ} $$

Short answer

The reference angle is 30°.

Step-by-step solution

Identify the Quadrant

Determine in which quadrant the given angle lies. Since 210° is between 180° and 270°, it is in the third quadrant.

Use Reference Angle Formula for the Third Quadrant

In the third quadrant, the reference angle is found by subtracting 180° from the given angle. So, compute 210° - 180°.

Perform the Calculation

Calculate the reference angle: 210° - 180° = 30°.

Key concepts

Reference Angles

A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.
Reference angles are always between 0° and 90°, inclusive. They help simplify the calculation of trigonometric functions for any angle.
The reference angle can be found for any given angle, whether it is positive or negative, by following specific rules depending on the quadrant in which the angle is located.
For instance, if we have an angle in the fourth quadrant, its reference angle would be calculated by subtracting it from 360°. If the angle was, say, 300°, the reference angle would be: 360° - 300° = 60°. Reference angles are great tools because they allow us to use familiar trigonometric values even for large or negative angles.

Trigonometric Quadrants

The coordinate plane is divided into four quadrants. Understanding which quadrant an angle lies in helps in calculating its reference angle.
Here's a quick rundown of the quadrants and the general range of angles in each:

• First Quadrant: 0° to 90° (angles are positive)
• Second Quadrant: 90° to 180° (angles are positive)
• Third Quadrant: 180° to 270° (angles are positive)
• Fourth Quadrant: 270° to 360° (angles are positive)

For example, an angle of 210° is in the third quadrant because it falls between 180° and 270°. Knowing this helps us quickly determine which formula to use for finding the reference angle.

Angle Subtraction

Once you know the quadrant an angle falls in, you can then use angle subtraction to find the reference angle.
Each quadrant has its own formula for this:

• First Quadrant: The reference angle is the angle itself.
• Second Quadrant: The reference angle is calculated as 180° - angle.
• Third Quadrant: The reference angle is calculated as angle - 180°.
• Fourth Quadrant: The reference angle is calculated as 360° - angle.

In the earlier example, 210° is in the third quadrant. The reference angle is found by subtracting 180° from 210°:
\[ 210^\text{°} - 180^\text{°} = 30^\text{°} \]
This angle (30°) is now easy to work with in terms of sine, cosine, and tangent functions.