Problem 75 - 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. $$\frac{29 \pi}{3}$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle is \(\frac{5\pi}{3}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. This means they look the same when drawn on a coordinate plane. To find a coterminal angle of a given angle, you can either add or subtract full rotations (360 degrees or \(2\pi\) radians for angles in radians).

Step 2: Identify Full Rotations in Radians

Since the given angle \(\frac{29\pi}{3}\) is in radians, a full rotation in radians is \(2\pi\). Determine how many full rotations (\(2\pi\)) are contained within the given angle. Each full rotation is equivalent to \(\frac{6\pi}{3}\).

Step 3: Calculate Rotations

Divide \(\frac{29\pi}{3}\) by \(\frac{6\pi}{3}\) to find the number of full rotations. The calculation is: \(\frac{29\pi}{3} \div \frac{6\pi}{3}\). This simplifies to \(\frac{29}{6}\).

Step 4: Determine the Whole Rotations

The quotient \(\frac{29}{6}\) can be divided to find the whole number of complete rotations. \(\frac{29}{6} = 4.833\ldots\), which means there are 4 full rotations.

Step 5: Subtract Full Rotations

Subtract these 4 full rotations to find the smallest positive coterminal angle. Calculate: \(\frac{29\pi}{3} - 4 \times 2\pi\). This simplifies to \(\frac{29\pi}{3} - \frac{24\pi}{3} = \frac{5\pi}{3}\).

Step 6: Verify the Smaller Angle

The angle \(\frac{5\pi}{3}\) is less than \(2\pi\), confirming it is the smallest positive coterminal angle. Since \(0 < \frac{5\pi}{3} < 2\pi\), it meets the criteria for being the smallest positive measure.

Key Concepts

Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which split a circle into 360 equal parts, radians use the circle's radius as the measuring stick. When you measure an angle in radians, you are essentially calculating how many radii you would "unwrap" around the circle to cover the arc of the angle.
Radians simplify many mathematical equations, especially in calculus and trigonometry. One full circle equates to an angle of \(2\pi\) radians. This simplification comes from the circumference of a circle being \(2\pi\) times the radius. When working with trigonometric functions or periodic functions, radians are commonly preferred over degrees as they align directly with the properties of these functions.

Full Rotation

A full rotation refers to a complete turn around a circle, resulting in the object or angle returning to its original position. In terms of measurement, one full rotation can be expressed in degrees or radians.

• In degrees, a full rotation equals 360 degrees.
• In radians, a full rotation is \(2\pi\) radians.

The concept of a full rotation is crucial when dealing with coterminal angles. Angles that differ by any whole number of full rotations are considered coterminal because they end at the same position on the coordinate plane. When working on problems involving coterminal angles, you often add or subtract full rotations to find equivalent angles in a desired measure, such as the smallest positive angle.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane used to graph mathematical concepts. It is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.
Angles are often represented on the coordinate plane, particularly in trigonometry. In this context, the angle originates from the intersection of the two axes, known as the origin. The initial side of the angle usually lies along the positive x-axis, while the terminal side is determined by the rotation that defines the angle. Visualizing angles on a coordinate plane can aid in understanding concepts like coterminal angles, as these visualizations highlight when different angles share the same terminal side.

Angle Measure

Angle measure is a fundamental concept that describes the size of an angle. It can be calculated in different units, most commonly degrees or radians.

• Degrees divide a circle into 360 equal parts.
• Radians are based on the circumference of the circle, with \(2\pi\) radians making up a full circle.

To gain a deeper comprehension of angles, it's important to understand both systems. Solving problems often requires converting between degrees and radians depending on the context. For instance, finding coterminal angles can involve switching between these units when necessary. Identifying the measure of an angle helps to determine positions and lengths in geometric shapes such as circles and polygons, making it fundamental to fields like trigonometry and geometry.