Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-330^{\circ}$$

Step-by-Step Solution

Answer

The reference angle \(\theta^{\prime}\) is 30 degrees.

Step 1: Convert to Positive Angle

Given that \(\theta = -330^{\circ}\), to convert this angle to a positive angle, we add 360 degrees to it: \(\theta = -330^{\circ} + 360^{\circ} = 30^{\circ}\)

Step 2: Determine the Reference Angle

The reference angle \(\theta^{\prime}\) for an angle in standard position is always positive and measures less than or equal to 90 degrees. Since \(\theta = 30^{\circ}\), which is already less than 90°, \(\theta^{\prime} = \theta\)

Step 3: Sketch the Angle

On a unit circle, plot the angle \(\theta = 30^{\circ}\) in standard position by rotating 30 degrees in the counter-clockwise direction from the positive x-axis. Label \(\theta^{\prime}\) as such.

Key Concepts

Positive Angles

When studying angles, understanding positive angles is essential. A positive angle is typically defined by its direction of rotation. If you start from the positive x-axis and rotate counterclockwise, you're creating a positive angle. This method is standard in mathematics and helps establish consistent directionality when working with angles on the unit circle.
For example:

• A rotation of 30 degrees counterclockwise from the positive x-axis is a positive angle of 30 degrees.
• Even if you initially had a negative angle, like -330 degrees in the original exercise, adding 360 degrees to it will convert it into a positive angle, resulting in 30 degrees.

The positive angle can easily align with our natural sense of positioning in mathematics, linking seamlessly into concepts like the unit circle.

Special Angles

Special angles are those that have notable trigonometric values, often used in trigonometry for their simplified sine, cosine, and tangent ratios. These angles typically include 30°, 45°, 60°, and their multiples. In our context, a special angle like 30° simplifies calculations and is easily handled in trigonometric functions.
Special angles are powerful tools in problem-solving because:

• They map neatly onto the unit circle, allowing for quick identification of sine and cosine values.
• They are integral to identifying reference angles, as their measurements rarely change across different quadrants.

For instance, when the original problem presented -330°, it was converted to its equivalent positive angle, 30°, which is considered a special angle with well-known sine and cosine values.

Unit Circle

The unit circle is a fundamental concept in trigonometry used to define angles and trigonometric functions. This circle has a radius of 1 and is centered at the origin of the coordinate plane. It is pivotal in connecting angle measures with coordinates.The unit circle allows us to:

• Understand how angles correlate to points: Each angle maps to a unique point (\(x, y\)) on the circle, where \(x\) and \(y\) represent the cosine and sine of the angle, respectively.
• Visualize special angles: Angles such as 30°, 45°, and 60° are easily plotted on the unit circle, showcasing their trig values at specific coordinates.
• Establish the reference angle: The process involves locating where an angle lies in the circle and determining the acute angle it forms with the x-axis.

In solving exercises, like the one you're tackling, the unit circle aids in sketching angles accurately and identifying reference angles that are crucial for calculations.