Problem 19 - Contemporary Precalculus

Question

Find the radian measure of an angle in standard position that has measure between 0 and \(2 \pi\) and is coterminal with the angle in standard position whose measure is given. $$-\pi / 3$$

Step-by-Step Solution

Answer

Answer: The coterminal angle with a measure between 0 and 2π for an angle in standard position with a measure of -π/3 is 5π/3.

Step 1: Identify the given angle

The angle in standard position given is: $$-\pi / 3$$

Step 2: Add multiples of \(2\pi\) to the given angle

Start by adding one full rotation (\(2\pi\)) to the given angle: $$-\pi / 3 + 2\pi $$

Step 3: Simplify the expression

To find the coterminal angle, simplify the expression from Step 2: $$-\pi/3 + 2\pi = (-\pi + 6\pi) / 3 = 5\pi/3$$

Step 4: Check if the simplified angle is within the desired range

The simplified angle is \(5\pi/3\). Check if it's within the range of \(0\) and \(2\pi\): $$0 \leq 5\pi/3 \leq 2\pi$$ This inequality is true, so the angle \(5\pi/3\) is the coterminal angle with the given angle \(-\pi/3\) and is within the range of \(0\) and \(2\pi\).

Key Concepts

Coterminal Angles

Coterminal angles are angles that share a terminal side when drawn in standard position on a coordinate plane. This means they point in the same direction, even though they might start from a different initial angle.
These angles are found by adding or subtracting full rotations, which are measured in radians or degrees.

• In radians, a full rotation around a circle is equivalent to adding or subtracting \(2\pi\).
• In degrees, you'd add or subtract 360 degrees.

When given an angle, such as \(-\pi/3\), you can find a coterminal angle by adding or subtracting multiples of \(2\pi\) until your angle is within the desired range.
For instance, the angle \(-\pi/3 + 2\pi = 5\pi/3\), meaning \(5\pi/3\) is a coterminal angle within the \([0, 2\pi]\) range.
By understanding how to find angles using these full-circle principles, identifying coterminal angles becomes a straightforward task.

Standard Position

In trigonometry, an angle is said to be in standard position if its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis.
This starting point simplifies the process of defining and comparing angles.

• The initial side is fixed and does not move; it's always where the angle starts.
• The terminal side is where the angle "ends" as it rotates from the initial side.

The importance of standard position lies in its universality and consistency, which allows us to categorize and easily find coterminal angles.
Thus, when working with angles in standard position, like \(-\pi/3\), you can confidently use the origin and the x-axis as your reference points to define the angle and its respective coterminal angles.
Remembering these simple points can help you better visualize and solve problems related to angles in trigonometry.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent provide a connection between the angles and the ratios of the sides of a right triangle.
Each function has a specific meaning when an angle is in standard position on the unit circle:

• Sine (\(\sin\)): Y-coordinate of the point on the unit circle.
• Cosine (\(\cos\)): X-coordinate of the point on the unit circle.
• Tangent (\(\tan\)): Ratio \(\frac{\text{Y-coordinate}}{\text{X-coordinate}}\).

Understanding these functions aids in calculating the values of angles like \(5\pi/3\), as we know the corresponding coordinates of this angle on the unit circle.
For example, knowing that \(\sin(5\pi/3)\) and \(\cos(5\pi/3)\) are based on these coordinates assists in solving problems involving angle measures, whether in radians or degrees.
So, these functions not only express these relationships but also become tools for deeper exploration and learning within trigonometry.