Question

Find the reference angle of each angle. $$ 120^{\circ} $$

Short answer

The reference angle is \(60^{\text{∘}}\).

Step-by-step solution

Identify the Quadrant

The first step is to determine in which quadrant the given angle lies. The angle given is \(120^{\text{∘}}\), which lies in the second quadrant (since \(90^{\text{∘}} < 120^{\text{∘}} < 180^{\text{∘}}\)).

Calculate the Reference Angle

To find the reference angle in the second quadrant, use the formula: \[ \theta_{\text{ref}} = 180^{\text{∘}} - \theta \ \text{where } \theta = 120^{\text{∘}} \]Thus: \[ \theta_{\text{ref}} = 180^{\text{∘}} - 120^{\text{∘}} = 60^{\text{∘}} \]

Key concepts

Quadrant Identification

Understanding in which quadrant an angle lies is essential in trigonometry. Each quadrant in the coordinate system helps us know the nature of the trigonometric functions of angles.

The coordinate system has four quadrants:

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

When given an angle like 120°, you should first figure out its position in the coordinate system. Since 120° lies between 90° and 180°, it is in the Second Quadrant.

In the Second Quadrant, the reference angle is always subtracted from 180°. This step is crucial as different quadrants have different rules for calculating reference angles.

Trigonometry

Trigonometry deals with the relationships between the angles and the sides of triangles. In this context, understanding the reference angle helps simplify problem-solving tasks involving trigonometric functions.

Key terms include:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Reference Angle

Using the reference angle can help you easily find trigonometric values. It serves as a simplified measure since it is the acute angle (less than 90°) formed by the terminal side of the angle and the x-axis. This approach is beneficial as it maintains the values within the more easily manageable range of 0° to 90°.

Angle Measurement

Angles can be measured in degrees (°) or radians. For most basic trigonometry problems, degrees are used. Understanding how to calculate reference angles involves straightforward subtraction.

With an angle of 120°, we already determined it lies in the Second Quadrant. Here, to find the reference angle, use the subtraction formula:

\[ \theta_{\text{ref}} = 180^{\text{∘}} - \theta \]
\[ \theta_{\text{ref}} = 180^{\text{∘}} - 120^{\text{∘}} = 60^{\text{∘}} \]

Thus, the reference angle is 60°. This value simplifies our trigonometric computations significantly. Using this method ensures consistency and accuracy in solving trigonometric problems.