Problem 44 - Algebra 2

Question

Find a positive and a negative coterminal angle for the given angle. $$ -57^{\circ} $$

Step-by-Step Solution

Answer

The positive coterminal angle to -57 degrees is 303 degrees and the negative coterminal angle is -417 degrees.

Step 1: Find the positive coterminal angle

Start by adding 360 degrees to the given angle. This is done because we are looking for a positive coterminal angle. \( -57^{\circ} + 360^{\circ} = 303^{\circ}\). Therefore, 303 degrees is a positive coterminal angle to -57 degrees.

Step 2: Find the negative coterminal angle

Now, subtract 360 degrees from the given angle. This gives us a negative coterminal angle. \( -57^{\circ} - 360 ^{\circ} = -417 ^{\circ}\). Therefore, -417 degrees is a negative coterminal angle to -57 degrees.

Key Concepts

Angle Measurement

When we talk about angles, we are describing the space between two intersecting lines or surfaces at or close to the point where they meet. It is like measuring the 'openness' between two lines coming from a point. Angles are commonly measured in degrees, which is a unit to express this measurement.

One full rotation around a circle is divided into 360 equal parts, making each part 1 degree (written as \(1^{\circ}\)). This system of angle measurement makes it straightforward to understand and communicate how open or closed an angle might be.

• If an angle completes a full circle, it measures \(360^{\circ}\).
• A right angle, like the corner of a piece of paper, measures \(90^{\circ}\).
• A straight angle, which looks like a straight line, measures \(180^{\circ}\).

By using degrees, we provide a simple and consistent way to discuss angles, whether they are small or wrap around a circle several times.

Positive and Negative Angles

Angles can be positive or negative, providing flexibility in how we measure directions. Positive angles are measured counterclockwise from the positive x-axis, and this is the traditional way we measure angles.

However, we can rotate in the opposite direction, giving us negative angles. Negative angles are measured clockwise from the same starting line. By using both positive and negative measurements, we can describe rotations in multiple ways.

• A positive angle has a measure of, for example, \(57^{\circ}\), moving counterclockwise.
• A negative angle moves clockwise and might be \(-57^{\circ}\).

Each positive angle has an infinite number of coterminal angles, including both positive and negative values. This flexibility is especially useful in applications like navigation, where direction is crucial.

Degrees in a Circle

The concept of degrees in a circle is fundamental to understanding angles and rotations. A circle is a closed curve with every point equidistant from the center.

To express how much of a circle's rotation an angle represents, we divide the circle into 360 degrees. This division is likely because it provides a number that is divisible by many factors (2, 3, 4, 5, 6, etc.), making calculations within circles easier and versatile.

• A full circle is \(360^{\circ}\), symbolizing full rotation.
• Half a circle, or a diameter extent, represents \(180^{\circ}\).
• Just a quarter turn will make \(90^{\circ}\).

Understanding these divisions helps with mapping angles and finding coterminal angles where necessary, like in the exercise of finding coterminal angles for \(-57^{\circ}\). You add or subtract \(360^{\circ}\) to find different positions in the full-circle rotation.