Defining the Radian in Trigonometry
A person's introduction to angles is usually in terms of degrees. You probably have a good idea of what a 45-degree angle looks like. And even most middle-school students know that a triangle consists of 180 degrees. But most of the scientific community uses radians to measure angles. Why change to radians?
Early mathematicians decided on the size of a degree based on divisions of a full circle. A degree is the same as a slice of 1/360 of a circle. No one knows for sure how the choice of 360 degrees in a circle came to be adopted. In any case, 360 is a wonderful number because it can be divide evenly by so many other numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. The early measures of time and distance relied on having convenient numbers to work with.
A radian, on the other hand, isn't quite as nice as far as being a whole number. Radians probably were developed because mathematicians wanted to relate the angle measure more to the radius or size of the circle.
A radian is much bigger than a degree. A circle has 2π radians (a little more than six radians). A radian is almost 1/6 of a circle — it's a little more than 57 degrees.
Relating to a circle
The big advantage of using radians is that they're the natural measure for dividing up circles. If you took the radius of a circle and bent it into an arc that lies along the circumference of the circle, you would need a little more than six of them to go all the way around the circle. This fact is true of all circles.
An angle measuring 1 radian would go through the two ends of one of those radii on the circle. The circumference of any circle is always a little more than three times the diameter of that circle — π times the diameter, to be exact. Another way of saying this is 2π times the radius. That number may seem nicer and more civilized than the big number 360, but the disadvantage is that π doesn't have an exact decimal value. Saying 2π radians (which is equal to 360 degrees) means that each circle has about 6.28 radians. Even though radians are the natural measure and always relate to the radius and diameter, the decimal values get a bit messy.
Each of these measures has its own place. Measuring angles in degrees is easier, but measuring angles in radians is preferable when doing computations. The radian is more exact because the radius, circumference, or area of the circle is involved. Even though π doesn't have an exact decimal value, when you use multiples of π in answers, they're exactly right.
The favorite or most used angles are those that are multiples of 15 degrees, such as 30, 45, 60, and 90 degrees. As luck would have it, converting those angles into radian measurements creates some nice, easy numbers to work with:
- A 30-degree angle is equivalent to π/6 radians.
- A 45-degree angle is equivalent to π/4 radians.
- A 60-degree angle is equivalent to π/3 radians.
- A 90-degree angle is equivalent to π/2 radians.
Radian measures with denominators of 2, 3, 4, and 6 are used most frequently.