Radian and Degree Measure

Trigonometry

Trigonometry is the relationship of a triangles sides and angles.

(Trigon) - Triangle

(ometry) - measure

Trigonometry - The measurement of triangles

Trigonometry is used to measure triangles, but more specifically, right triangles.


Angles

An angle is defined as two rays with a common endpoint.

In the diagram above, an angle is formed by the rays P and R with endpoint Q.

With an angle, what is being measured is the amount of rotation between the two rays.

Angles can be measured in different units such as degrees, radians, grads, and revolutions.

Just to clear up a common misconception, ANGLES CAN BE NEGATIVE, they are vectors and their sign dictates direction of ray rotation.

Now that we know that angles are vectors, we need to define a start and end of the angle. The start of an angle will be one ray, called the Initial Side, and the end will be called the Terminal Side.

In this case theta is the name of the angle.

What if the angle was not labeled, how would you know which side was the terminal side?

This is why there is something called standard position, and it dictates the initial and terminal sides.

Standard Position - The initial side is on the x-axis and the vertex is at the origin.

There are four key words that are needed to know about a set of angles.

Congruent - The angles have the same measure.

Complementary - the sum of the angles is 90 degrees.

Supplementary - the sum of angles is 180 degrees.

Co-Terminal - The angles have the same terminal side. Ex. 40, 400, -320 degrees.

Here is an example of Co-Terminal Angles

All of the above angles have the same terminal side, which makes them Co-Terminal.

Arc Length

Arc length on a circle is usually denoted as "S" on a circle.

In the diagram above, an arc is formed by angle theta.

The formula to find the arc length when using degrees is as follows:

The formula to find arc length when using radians and ONLY RADIANS is as follows:

Conversions

Often in trigonometry, it will be necessary to convert Radians to Degrees and so on. The best way to do that is by using a conversion factor.

If I need 30 degrees in Radians, I will use a conversion factor to find it.