Question

Find the reference angle of each angle. $$ 440^{\circ} $$

Short answer

80°

Step-by-step solution

Understand Reference Angle

The reference angle is the smallest angle between the given angle and the x-axis. It is always between 0° and 90°.

Reduce the Angle

To find the reference angle for an angle greater than 360°, subtract 360° to bring the angle within one full rotation. For 440°: \( 440^{\text{\textdegree}} - 360^{\text{\textdegree}} = 80^{\text{\textdegree}} \)

Determine the Reference Angle

Since the reduced angle from Step 2 is already within 0° and 90°, the reference angle is simply \( 80^{\text{\textdegree}} \).

Key concepts

angle reduction

When working with angles in trigonometry, sometimes we encounter angles that are larger than 360°. This means they have completed more than one full rotation around a circle.

To simplify these larger angles, we use a process called angle reduction. The goal is to reduce the angle until it falls within the first full rotation, between 0° and 360°. We achieve this by subtracting 360° for every complete rotation the angle has made.

For example, in the exercise given with a 440° angle:

• Subtract 360° from 440°
• This gives us 440° - 360° = 80°

Now the angle is reduced to 80°, which is within the first full rotation.

trigonometric angles

Trigonometric functions (sine, cosine, tangent, etc.) often deal with angles measured from the x-axis. These angles measure the rotation from the initial side (positive x-axis) to the terminal side of the angle.

These angles can be positive or negative:

• Positive angles are measured counterclockwise from the x-axis.
• Negative angles are measured clockwise from the x-axis.

When we talk about trigonometric angles, it’s important to understand how they repeat every 360°. That’s why angle reduction is useful: it allows us to find an equivalent angle within the first 360° rotation, making it easier to work with trigonometric functions.

x-axis reference

The x-axis serves as a primary reference line in the coordinate system for measuring angles. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It always lies between 0° and 90°.

This is because the reference angle helps to find the corresponding values of trigonometric functions using the unit circle.

For example:

• If you have an angle of 80° (already reduced from 440°), and it's in the first quadrant, the reference angle is 80° itself.
• If it were in another quadrant, you would determine the smallest angle to the x-axis from the terminal side.

The reference angle simplifies complex trigonometric calculations by standardizing them to the first quadrant.