Question

Find the reference angle of each angle. $$ 320^{\circ} $$

Short answer

The reference angle is 40°.

Step-by-step solution

Identify the Quadrant

First, determine in which quadrant the given angle lies. The angle provided is 320°. Since it is between 270° and 360°, it lies in the fourth quadrant.

Use the Reference Angle Formula

For an angle in the fourth quadrant, the reference angle can be found using the formula: \[\text{Reference angle} = 360^{\text{o}} - \theta\] where \theta\ is the given angle.

Substitute and Calculate

Substitute the given angle into the reference angle formula: \[ \text{Reference angle} = 360^{\text{o}} - 320^{\text{o}} = 40^{\text{o}} \]

Key concepts

quadrants

In trigonometry, the coordinate plane is divided into four parts called quadrants. Each quadrant houses angles and gives insight into their properties. Understanding quadrants makes it easier to grasp angle behaviors.

• First Quadrant: 0° to 90°
• Second Quadrant: 90° to 180°
• Third Quadrant: 180° to 270°
• Fourth Quadrant: 270° to 360°

Knowing these can help you determine the reference angle, which is the smallest angle formed with the x-axis. For example, our angle of 320° falls into the fourth quadrant since it lies between 270° and 360°.

angle measurement

An angle's measurement can be determined in degrees. Degrees are a way to measure how far you’ve rotated from one point to another. In a circle, which is 360°, we can split this into four right angles, each being 90°. To visualize, picture a full circle divided into four equal parts. Each part represents one of the quadrants. Using degrees, we can pinpoint exactly where an angle is on a circle. By knowing the degrees, you can identify the quadrant it belongs to—essential in problems like finding reference angles.

trigonometry formulas

Trigonometry relies on specific formulas to solve angles and sides of triangles. For finding reference angles, different quadrants use distinct formulas. Here's a brief overview:

• First Quadrant: Reference angle = θ
• Second Quadrant: Reference angle = 180° - θ
• Third Quadrant: Reference angle = θ - 180°
• Fourth Quadrant: Reference angle = 360° - θ

The formula for our 320° angle, lying in the fourth quadrant, becomes: \[Reference \, angle = 360^{\circ} - 320^{\circ} = 40^{\circ}\] Using these formulas helps simplify otherwise complex angle calculations.

degrees

Degrees are a crucial unit of measurement in trigonometry. They help us communicate and understand angles precisely. One complete rotation around a circle is 360°, broken down into smaller parts.

• 1° = 1/360 of a full rotation
• 90° = right angle
• 180° = straight line

In trigonometry, it’s common to also consider angles beyond 360°, by effectively adding full circles. Degrees help us break down an angle to its smallest component, making it easier to find key values like reference angles. For example, the angle 320° can be referenced by subtracting from 360°, resulting in a much smaller, easier to understand, 40° reference angle.