In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -𝜋/40

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Understand that two angles are coterminal if they differ by an integer multiple of \(2\pi\). This means we can add or subtract \(2\pi\) to the given angle to find coterminal angles.

The given angle is \(-\frac{\pi}{40}\). Since it is negative, to find a positive coterminal angle less than or equal to \(2\pi\), add \(2\pi\) to it.

Write the expression for the coterminal angle: \(-\frac{\pi}{40} + 2\pi\).

To add these, express \(2\pi\) with a denominator of 40: \(2\pi = \frac{80\pi}{40}\). So the sum becomes \(-\frac{\pi}{40} + \frac{80\pi}{40} = \frac{79\pi}{40}\).

Verify that \(\frac{79\pi}{40}\) is positive and less than or equal to \(2\pi\) to confirm it is the desired coterminal angle.

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주요 개념

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Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations of 2π radians. To find a coterminal angle, you add or subtract multiples of 2π until the angle lies within the desired range, such as between 0 and 2π.

Angle Measurement in Radians

Angles can be measured in radians, where 2π radians equal 360 degrees. Understanding radian measure is essential for working with trigonometric functions and converting between degrees and radians when necessary.

Positive Angle Determination

Finding a positive angle less than or equal to 2π involves adjusting the given angle by adding 2π repeatedly if it is negative. This ensures the angle is expressed within the standard interval for trigonometric analysis.