Overview: What Is Radian Measure?
The radian measure is another unit of angle measurement. An angle has the measurement of one radian if the length of the arc it intercepts is equal to the measurement of one radius. The radian measurement has many applications in mathematics and physics, where the more familiar measurement of angles by degrees is not as useful.
How Many Radians Are There in a Circle?
Suppose a circle has a radius of 3 units, and an arc is drawn that intercepts 6 units on the circumference. The number of radians is 6/3 or 2. If the arc was all the way around the circumference of the circle (C), then the arc would be equal to C. Remembering how the measurement of the length of the circumference of the circle is derived as C=2πr, the number of radians will equal 2πr/r or 2π.
Converting Degrees to Radians and Radians to Degrees
If 2π radians =360o, then π radians =180o, by dividing both sides of the equation by 2. Therefore 1 radian will equal 180o/π, or about 57.2958 degrees, and 1 degree will equal π/180, or about 0.0175 radians. The angle in degrees will equal the angle in radians times 57.2958. The angle in radians will equal the angle in degrees times 0.0175.
Relationship of Radian Measure to Trigonometric Functions
There are many uses of sine, cosine, and tangent in mathematics beyond the measurement of angles. Formulas used in math analysis and calculus measure angles in radians rather than degrees. They are based upon functions that are simplified into radians. For example, the relationships between exponential functions and sine and cosine functions are expressed in radian units rather than degrees.
Applications of Radian Measure
Many measurements in physics, such as the phase difference between waves, are measured in radians. Military applications of targeting, as well as telescopic sights, use thousandths of radians, because it is more convenient. The measurement of small angles in astronomy is also often done in decimal fractions of radians.
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