Converting Radians to Degrees

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Degrees and radians are two different units used to measure angles.

A degree (denoted by the symbol °) is defined as 1/360 of a full circle. This makes it an easy to understand measure for angles, as a full circle is 360°, a half circle (or a right angle) 180°, and a quarter circle 90°. Degrees are the standard unit for measuring angles in everyday life and in most areas of mathematics.

Radians (rad), on the other hand, are the standard unit for measuring angles in higher mathematics and in many scientific fields, such as physics. A radian is defined as the angle subtended by the radius of a circle, which means that a full circle is 2π radians, a half circle is π radians, and a quarter circle is π/2 radians.

While it may not seem so at first glance, radians are a very natural way to measure angles, and many mathematical formulas become simpler and more elegant when expressed in terms of radians. This is especially noticeable in calculus and trigonometry, where many identities and formulas assume that angles are measured in radians.

There is a simple conversion between degrees and radians: to convert degrees to radians, multiply by π/180, and to convert radians to degrees, multiply by 180/π.

rad = degrees * (π/180)

degrees = rad * (180/π)

In this illustration, the red line is the radius (r) of the circle, the blue arc shows the radian angle (θ rad), and the black line at the edge of the circle shows the equivalent degree angle (θ degrees). As we can see, degrees and radians are different ways of measuring the size of an angle.

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