Problem 70 - Precalculus

Question

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$379^{\circ}$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle with \(379^{\circ}\) is \(19^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles, we either add or subtract full rotations of \(360^{\circ}\).

Step 2: Finding Coterminal Angles

For the angle \(379^{\circ}\), we need to subtract \(360^{\circ}\) to find an equivalent angle that's between \(0^{\circ}\) and \(360^{\circ}\). We calculate \(379^{\circ} - 360^{\circ} = 19^{\circ}\).

Step 3: Verify the Smallest Positive Angle

Ensure that the angle \(19^{\circ}\) is indeed between \(0^{\circ}\) and \(360^{\circ}\). Since \(19^{\circ}\) is within this range, it's the smallest positive coterminal angle for \(379^{\circ}\).

Key Concepts

Angle Measures

Angle measures are a fundamental part of understanding everything that involves rotation, circles, and direction. When we talk about angles, we are considering the space between two intersecting lines. This space is measured in degrees or radians, depending on the context and use.
There are many applications of angle measures in mathematics, particularly in geometry and trigonometry. Angles can be acute, right, obtuse, or even reflex, based on how open or closed they are:

• An acute angle is less than 90 degrees.
• A right angle is exactly 90 degrees.
• An obtuse angle is greater than 90 degrees but less than 180 degrees.
• A reflex angle is greater than 180 degrees.

Understanding angle measures is crucial for working with shapes and for moving into more complex topics like trigonometry. Properly measuring and identifying angles helps in solving many practical and theoretical problems.

Degrees

Degrees are a common unit of measure for angles, and they are often used in daily applications as well as in academic settings. A full circle is divided into 360 degrees (\[360^{\circ}\]), which is a standard that dates back to ancient Babylon. This division is convenient because many numbers divide evenly into 360, allowing for easier calculations in various applications.
When you work with degrees:

• 0 degrees represents no rotation.
• 90 degrees forms a right angle.
• 180 degrees is a straight line.
• 360 degrees completes a full circle.

Degrees travel around in a circular path and can continue indefinitely. When you have angles greater than 360 degrees, like \[379^{\circ}\], you can find equivalent or coterminal angles by subtracting or adding multiples of \[360^{\circ}\] until you reach a simple, easily understood angle between \[0^{\circ}\] and \[360^{\circ}\].

Trigonometry

Trigonometry is a branch of mathematics that deals with relationships between angles and sides of triangles. It's a broad field that extends far beyond basic triangles and finds applications in physics, engineering, and even economics.
At its core, trigonometry relies on understanding angles and how they interact with circular functions like sine, cosine, and tangent. Knowing the measures of angles, as well as fitting them into a 360-degree circle, helps in comprehensively understanding trigonometric functions.
In trigonometry, angles are often expressed in degrees or radians. While we've focused on degrees here, it's important to remember that radians are another form of measurement that deals with the arc length of a circle. For trigonometric identities and equations, being comfortable with both measurement scales can be beneficial.

• Sine of an angle measures the opposite side over the hypotenuse in a right triangle.
• Cosine measures the adjacent side over the hypotenuse.
• Tangent measures the opposite side over the adjacent side.

Trigonometry and its principles enable us to explore and solve many complex problems involving angles and their measures.