Question

Find the reference angle of each angle. $$ 490^{\circ} $$

Short answer

The reference angle is 50°.

Step-by-step solution

Understand reference angles

A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It is always measured in the positive direction.

Find the coterminal angle in the range [0^{\text{ \textdegree}}, 360^{\text{ \textdegree}}]

To find the coterminal angle, subtract multiples of 360° from 490° until the result is within the range [0°, 360°]. 490° - 360° = 130° .

Determine the reference angle

Since 130° is in the 2nd quadrant, the reference angle is calculated by subtracting the angle from 180°: 180° - 130° = 50°.

Key concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. This means that even though they may have different measures, they essentially point in the same direction.
The simplest way to find a coterminal angle is by adding or subtracting multiples of 360° because a full circle is 360°. For example, starting with an angle of 490°:

• 490° - 360° leads to 130°
• We could also add 360° to a negative angle, like -30°, and get 330° because -30° + 360° = 330°

The key is that adding or subtracting 360° from an angle gives you another angle that looks the same on a unit circle.

Quadrants

The coordinate plane is divided into four sections known as quadrants, each with specific characteristics. This system helps us identify the sign of the sine, cosine, and tangent functions based on the angle's position.

• First Quadrant: Angles from 0° to 90°, both sine and cosine are positive.
• Second Quadrant: Angles from 90° to 180°, sine is positive, cosine is negative.
• Third Quadrant: Angles from 180° to 270°, both sine and cosine are negative.
• Fourth Quadrant: Angles from 270° to 360°, sine is negative, cosine is positive.

In our example, 130° falls into the second quadrant, so we use this information to find the reference angle by subtracting from 180°.

Angle Measurement

Angle measurement can be in degrees or radians, but degrees are more common for basic trigonometry. Angles in standard position are measured from the positive x-axis counterclockwise.
Here's a quick overview:

• Full circle: 360°
• Straight angle: 180°
• Right angle: 90°

When measuring angles greater than 360° or negative angles, we often use coterminal angles to simplify. For instance, the angle 490° can be thought of as equivalent to 130°, by subtracting 360°. This helps us work with simpler values while acknowledging the angle's true measure.