In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle.- 38𝜋9

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First, understand that coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle, you can add or subtract full rotations (multiples of $2\pi$ radians) from the given angle.

Given the angle $\frac{38\pi}{9}$, we need to find a positive angle less than $2\pi$ that is coterminal with it.

Calculate the number of full rotations in the given angle by dividing it by $2\pi$: $\frac{38\pi}{9} \div 2\pi = \frac{38}{18} = \frac{19}{9}$. This means the angle is more than 2 full rotations.

Subtract the integer part of the division (which is 2 full rotations) from the given angle: $\frac{38\pi}{9} - 2 \times 2\pi = \frac{38\pi}{9} - \frac{36\pi}{9} = \frac{2\pi}{9}$.

The angle $\frac{2\pi}{9}$ is positive and less than $2\pi$, making it the coterminal angle we are looking for.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find a positive angle that is coterminal with a given angle, you can add or subtract multiples of 2π (or 360 degrees) until the angle falls within the desired range, typically between 0 and 2π.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One full rotation around a circle is 2π radians. Understanding how to convert between radians and degrees is essential for working with angles in trigonometry, especially when determining coterminal angles.

Modulo Operation

The modulo operation is used to find the remainder of a division. In the context of angles, it helps in determining coterminal angles by effectively 'wrapping' the angle around a full rotation. For example, to find a positive angle coterminal with 38π/9, you can use the modulo operation with 2π to simplify the angle into the desired range.

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