Problem 21 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ -10^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is 350°.

Step 1: Understanding Coterminal Angles

First, let's understand the concept of coterminal angles. Two angles are coterminal if they share the same initial and terminal sides. To find coterminal angles, you can add or subtract multiples of 360°, because a full rotation is 360°.

Step 2: Identifying the Initial Angle

The given angle is \(-10^{\circ}.\) This angle is already in degrees but is negative, which means it is measured in the clockwise direction from the positive x-axis.

Step 3: Finding a Positive Coterminal Angle

Since the angle is negative and we want the coterminal angle to be between 0° and 360°, we add 360° to \(-10^{\circ}\)to adjust it, \(-10° + 360° = 350°\).

Step 4: Checking the Range

Now, we check if \(350^{\circ}\) falls within the desired range \(0 \leq \theta < 360.\) Since 350° is within this range, it is a valid coterminal angle.

Key Concepts

Angle Measurement

Understanding how angles are measured is fundamental to grasping concepts like coterminal angles. Angles can be measured in a few different units, but the most common are degrees and radians. In geometry and trigonometry, degrees are widely used because they break down a circle into 360 equal parts, making calculations straightforward.

When measuring angles in degrees, a full circle is represented as 360 degrees. Angles are typically measured starting from the positive x-axis and can be measured in both clockwise and counterclockwise directions. Angles measured counterclockwise are positive, while those measured clockwise are negative. For example, a negative angle such as -10° indicates that the measurement started from the positive x-axis and rotated 10 degrees in the clockwise direction.

• Positive Angles: Measured counterclockwise from the positive x-axis.
• Negative Angles: Measured clockwise from the positive x-axis.

Understanding these basics can significantly aid in solving problems related to coterminal angles and ensuring you always know how an angle is positioned in relation to the principal axis.

Degrees

Degrees are one of the most familiar ways to measure angles in math courses. The degree system divides a full circle into 360 uniform parts or degrees. This division makes calculations simple and intuitive, especially when dealing with rotations, turning motions, or basic trigonometry.

In everyday contexts, you might encounter angles such as:

• Right Angle: 90 degrees.
• Straight Angle: 180 degrees.
• Full Circle: 360 degrees.

You can also have angles that are less than a full rotation, like -10°, which represents 10 degrees of rotation clockwise from the starting axis.

Being aware of how many degrees make up the angles and how to manipulate them by adding or subtracting 360 degrees helps in finding coterminal angles. By mastering the use of degrees, you open up a world of easier angle calculations and transformations in various mathematical problems.

Full Rotation

The concept of a full rotation is essential when working with angles, particularly in the context of coterminal angles. A full rotation is defined as a complete 360-degree spin around a central point, often visualized as a clock's hand moving back to where it started.

In mathematical terms, understanding full rotation is vital because it lays the groundwork for identifying coterminal angles. Since 360 degrees mark the start and end of a circle, adding or subtracting this amount gives you an angle pointing in the same direction but possibly with a different value.
For instance, starting with an angle like -10°, if you add 360° (a full rotation), you adjust the angle's position without changing its direction. This operation yields 350°, demonstrating that angles can differ by full rotations and still be essentially the same in terms of direction.

• Coterminal Angles: Angles that differ by full rotations.
• Adding 360°: Counteracts negative measurements to find positive coterminal angles.

Recognizing how full rotations interact with angle measurements is key to comprehending trigonometric problems, ensuring accurate calculations and a deeper understanding of angular movement.