3.4 Radian Measure of Angles

Hey everybody! Here is the lowdown on radian measures of angles!

The Main Concepts are...

The relationship between pi and a circle is that the circumference divided by 2 times the radius = pi. The radian is the length of the radius of the circle. The relationship between a radian and a circle is that pi is equal to 3.14...radians that wrapped around the circle, like this:

This graph is supposed to be on the UV axis, with the radius at (0,0).

Where the radians intersect the circle creates an angle with the center the circle. The angles would be same in all circle, for the radian is proportionate circumference. The angles will look like this:

Where 1, 2, and 3 stand for the angles created by the number of radians. The angles made by the radian will always have the same length in the arc as the legs of the angle, and can be described by the formula A1/R1 = A2/R2, because both equations are equal to 1. The angle the is created by the radian

subtends, or cuts off, a fragment of the circle which is the length of the radian.

The measure of a radian is always the arc length divided by the measure of the radius. Ex: if the arc were 3 units and the radius were 1.5 units, the radian would be 2.

If you know the radian measure of an angle, it is possible to express the measure that way, using (while in RADIAN mode) the small r (under 2nd, APPS).

When you know the angle measure in degrees and want to find the measure in radians, the formula is pi/180 degrees. In order to find the degree measure when you have the radian measure, the formula is 180 degrees/pi.

The most common radian measures are those of 30, 60, 90, 45, 180, and 360 degrees. The radians are exactly pi/6, pi/4, pi/3, pi/2, pi, and 2pi respectively. These number come from the number of degrees in pi, or 180, divided by the given degree. Ex: 180/30 = 6, which is in its corresponding radian formula. To find the number of full revolutions, simply divided 360 degrees by the given angle measure. Ex: 360/30 = 12, thus a radian with that measure goes around the circle 12 times evenly.

Example Problems!

1. find the exact radian measure of 45 degrees.

To solve this, use the formula of ( pi/180 ) x 45. The answer is pi/4.

2. Find the exact degree measure of 3pi/4.

To solve this, use the formula of ( 180/pi ) x (3pi/4). When you multiply it out, the formula is 540pi/4pi, which simplifies to 135 degrees.

3. Find the exact value of sin pi/3.

To solve this, use the change the radian measure of the angle to degrees, which is 60 degrees. Then solve like we have previously, using the 30, 60, 90 triangle.

If you need more info on this topic, check out http://aleph0.clarku.edu/~djoyce/java/trig/angle.html

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