Problem 10 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(210^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is \(30^{\circ}\).

Step 1: Understand What a Reference Angle Is

A reference angle is the smallest angle that a given angle makes with the x-axis, measured in a positive direction. For angles in different quadrants, the reference angle helps to find equivalent angles in the first quadrant.

Step 2: Determine Which Quadrant the Angle is in

The given angle is \(210^{\circ}\). An angle between \(180^{\circ}\) and \(270^{\circ}\) falls in the third quadrant. Thus, \(210^{\circ}\) is in the third quadrant.

Step 3: Calculate the Reference Angle

In the third quadrant, the reference angle \( \theta_{\text{ref}} \) is calculated by subtracting \(180^{\circ}\) from the angle. Thus, \( \theta_{\text{ref}} = 210^{\circ} - 180^{\circ} \).

Step 4: Perform the Subtraction

Perform the subtraction: \(210^{\circ} - 180^{\circ} = 30^{\circ}\). This results in a reference angle of \(30^{\circ}\).

Key Concepts

Understanding Trigonometric Quadrants

In trigonometry, the coordinate plane is divided into four quadrants, which help us determine the sign and value of trigonometric functions of angles. Each quadrant is defined by the range of angles it contains:

• First Quadrant (0° to 90°): All trigonometric functions are positive here.
• Second Quadrant (90° to 180°): Sine is positive, while cosine and tangent are negative.
• Third Quadrant (180° to 270°): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive, while sine and tangent are negative.

Understanding these quadrants is essential because they help predict the direction and sign of the trigonometric values. It is crucial for tasks like finding a reference angle, as it provides context on how angles relate to one another across quadrants.

The Art of Angle Measurement

Angles are measured in degrees or radians, with a full rotation around a circle being 360 degrees or \(2\pi\) radians. Measuring angles start at the positive x-axis and move around counter-clockwise. However, the notion of measuring angles can get tricky when it comes to large or negative angles, creating the need for simplification or conversion, such as finding equivalent angles.

When it comes to finding reference angles, understanding how angles correspond within each quadrant becomes significant. Reference angles simplify calculations by reducing complex angles to the smallest angle they make with the x-axis.

For instance, if you have an angle of 370°, you can subtract 360° (a full rotation) to simplify it to 10°, making calculations easier and linking it back to a reference angle.

Exploring Third Quadrant Angles

Angles in the third quadrant are those that range from 180° to 270°. These angles have special characteristics:

• They all have sine and cosine values that are negative. This is because they are located both below the x-axis and to the left of the y-axis.
• The tangent of third quadrant angles is positive, as the negatives of sine and cosine cancel each other out when divided.

To find the reference angle of a third quadrant angle like 210°, you simply subtract 180° from the given angle. This is because, in the third quadrant, angles are measured starting from the negative x-axis going counter-clockwise.

Therefore, for an angle such as 210°, subtracting 180° yields 30°, which is the reference angle. This helps in simplifying trigonometric calculations by using known values from the first quadrant.