Radian Measure of Angles

We start this section off by recalling that an angle AOB (denoted AOB) consists of two rays R₁ and R₂ with a common vertex O. An angle is often thought of as a rotation of the ray R₁ onto R₂ where R₁ is the initial side and R₂ is called the terminal side of the angle. A counterclockwise rotation is positive and a clockwise rotation is negative. The amount of rotation about the vertex is known as the measure of an angle. When it is rotated counterclockwise until it coincides with itself it is said to measure 360 degrees which is one revolution.

If a unit circle is drawn with the vertex of an angle at its center, then the measure of this angle in RADIANS is the length of the arc that subtends the angle. Take a look at the diagram to see what we mean.

As we saw in the circles section, the circumference of the circle of radius 1 is 2. Therefore one complete revolution has a measure of 2 radians. Also note that a straight angle has a measure radians, and a right angle has a measure /2 radians.

So what is the relationship between degrees and radians? It's easy to figure out. We know that a straight angle is 180 degrees and it has a measure of radians. Therefore 180^o = radians. Hence we get 1 rad = (180/)^o, and 1^o = (/180) rad. Let's look at some examples.

When an angle is drawn in the xy-plane with the vertex at the origin and its initial side on the positive x-axis, then it is said to be in standard position. If two angles in standard position are coinciding, then there are coterminal. Here is a diagram to help visualize this. To find a coterminal angle given another angle, all we must do is add a multiple of 360^o if the angle is measured in degrees or add a multiple of 2 if the angle is measured in radians. If the multiple is positive, then it is a positive coterminal angle and if it is a negative multiple then it is a negative coterminal angle.

We can figure out the arc length of a circle if we know the radius and we know the measure of the angle which the arc subtends. For instance, if the angle measures radians, the arc length s is s = r.

Note:

This is how we came up with the formula. Since the fraction of the circle contained by the arc is /2 and the circumference of the circle is given by 2r, we get the arc length
s = (/2)(2r) = r.

Here are a few examples.

A similar thing can be done if we know the area of a circle. Instead of finding the length of the arc, we find the area of the sector of the circle. This is given by the formula

A = /(2) * (r²) = r²/2.

Let's look at a couple of examples dealing with this idea.

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