Textbook Question
Find the reference angle for each angle.
205°
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3. Unit Circle
Reference Angles
3:23 minutesProblem 59
Textbook Question
Find the reference angle for each angle.
-25π/6
Verified step by step guidance1
Convert the given angle from radians to degrees by using the conversion factor \(180^\circ/\pi\).
Calculate the equivalent positive angle by adding \(2\pi\) (or \(360^\circ\)) until the angle is positive.
Determine the coterminal angle by finding the remainder when the positive angle is divided by \(2\pi\) (or \(360^\circ\)).
Identify the reference angle by finding the acute angle formed with the x-axis, which is the smallest angle between the terminal side of the angle and the x-axis.
If the coterminal angle is in the second or third quadrant, subtract it from \(\pi\) (or \(180^\circ\)); if in the fourth quadrant, subtract it from \(2\pi\) (or \(360^\circ\)).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reference Angle
The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always measured as a positive angle and is used to simplify the calculation of trigonometric functions. For angles in standard position, the reference angle can be found by subtracting or adding multiples of π/2 or π, depending on the quadrant in which the angle lies.
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Angle Measurement
Angles can be measured in degrees or radians, with radians being the standard unit in trigonometry. To convert between the two, one radian is approximately 57.3 degrees. Understanding how to work with angles in radians is crucial, especially when dealing with angles greater than 2π or less than 0, as is the case with negative angles.
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Coterminal Angles
Coterminal angles are angles that share the same terminal side but differ by a full rotation, which is 2π radians (or 360 degrees). To find a coterminal angle, you can add or subtract 2π from the given angle. This concept is particularly useful when working with angles that are outside the standard range of 0 to 2π, as it allows for simplification when finding reference angles.
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