Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-292^{\circ}$$

Step-by-Step Solution

Answer

The reference angle \(\theta^{\prime}\) of \(\theta = -292^{\circ}\) in standard position is \(\theta^{\prime} = 68^{\circ}\).

Step 1: Convert \(\theta\) into a positive angle

As a rotation of \(360^{\circ}\) actually adds nothing to the position, it is possible to add or remove full turns to \(\theta\) to transform it into an equivalent angle. Here, adding \(360^{\circ}\) to \(-292^{\circ}\) makes \(\theta\) equal to \(68^{\circ}\), a positive angle also on the standard form.

Step 2: Find \(\theta^{\prime}\)

Since \(68^{\circ}\) lies in the first quadrant where all reference angles are the same as the angles themselves, then \(\theta^{\prime} = 68^{\circ}\).

Key Concepts

Standard Position

When dealing with angles, it's important to understand the concept of standard position. This is where an angle's vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis. Any rotation determines this angle, moving counterclockwise for positive angles and clockwise for negative angles.

Understanding standard position helps in visualizing the angle's location and determining its reference angle. The reference angle is always a positive acute angle between the terminal side of the angle and the x-axis.

Angle Conversion

Angle conversion is a key step in finding reference angles, especially when you start with a negative angle. Negative angles rotate clockwise, which can make it challenging to visualize and calculate their reference angles.

To convert to a positive angle, you can add or subtract complete rotations (360°) until you obtain a positive angle. For example, with \( \theta = -292^\circ \), adding 360° gives \( 68^\circ \) as the equivalent positive angle. A positive angle is much easier to handle in calculations and visualizations.

First Quadrant

The first quadrant is a specific area on a coordinate plane, where both x and y values are positive. This lies between 0° and 90°, and angles in this quadrant are very straightforward to handle.

In the first quadrant, the reference angle is the same as the angle because it is already acute. For example, after converting \( \theta = -292^\circ \) to \( 68^\circ \), it falls into the first quadrant, making the reference angle also \( 68^\circ \).

Negative Angle Transformation

Turning a negative angle into a more manageable form involves understanding its transformation. Negative angles rotate in a clockwise direction on the coordinate plane. Visualizing this can be tricky, so converting them to positive angles simplifies the task.

By adding multiples of 360°, you convert a negative angle into a positive one. This allows you to locate its standard position easily. For instance, with \( \theta = -292^\circ \), the transformation results in a positive \( 68^\circ \). Such transformations help in easily identifying angle properties like the reference angle.