Angle Measurement

Radians and degrees are two ways of measuring the same thing.  Angle measurement can be calculated, in both degrees and radians, but in certain cases radians are more important.  There are 360 degrees in a circle and 2π radians in a circle.  We usually use fractions with pi when talking about radians, because it is actually easier to work with the fractions.  Pi radians translates to 180 degrees and half of that 90° = π/2 = 1.5707964 as a decimal equivalent.  Radians are just another form of measurement that can be used to scale things with larger form.  Anything that is measured in degrees can also be measured in radians.  One degree has a measure of 60 min, and 0.5 degrees is equivalent to 0.5 * 60 = 30 minutes, so 30 minutes is half of one degree.

Our counting and measuring system is based on the number 10, but the Babylonians used a base 60 number system.  The number 360 is totally arbitrary, it was chosen simply because the Babylonians preferred multiples of 60.  The circumference of a circle ‘C’ is equal to the length of the radius ‘r’ times 2π, thus C = 2πr.  Degrees let us work with integers (like 30o) instead of nasty ratios involving irrational numbers (like π/6) and they are easier for most measurements, but Trigonometry marked a turning point in math, and to navigate that terrain, you need a notion of angles that’s more natural, more fundamental, than slicing up the circle into an arbitrary number of pieces.  The number π, strange though it may seem, lies at the heart of mathematics.  The number 360 doesn’t fit well with pi and radians can take you places that degrees simply can’t.

The new way of teaching trigonometry works with the unit circle, which is a circle that has a radius of ‘one’ and because it is so simple, it has become the new method to learn about lengths and angles.  The polar coordinates (r, θ) of the point P, where r is the distance that P is from the origin O and θ (theta) is the angle between the lines Ox and OP, this is shown in the following diagram.

In the Polar Coordinate System, we go around the origin or the pole a certain distance out, and a certain angle from the positive x axis.  The ordered pairs, called polar coordinates, are in the form (r, θ), with r being the number of units from the origin or pole (if r > 0), like a radius of a circle, and θ being the angle (in degrees or radians) formed by the ray on the positive x axis (polar axis), going counter-clockwise.

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