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Episode 35 – A circle, angles, and a triangle.

In this post we switch gears again and present some math to help us calculate and understand some things.

It will be helpful to review Episode 17 Triangle & Complementary Angles and  Episode 24 Ratio and Proportion and Right Triangles before listening to this episode..

First I define a special type of circle.  The radius of the circle is one.   This has been chosen so it can be easy to scale up or scale down the circle (and later the triangle).

The radius is the distance from the center of the circle to the edge.  The diameter is the distance of a line going from one side of the circle to the other side while going through the center of the circle.  This means Diameter = 2 X radius.

The starting point of a circle is the 3 O’clock position and by definition a positive angle is moving around the circle counter-clockwise.    So +90 degrees would be the 12 O’clock position while the 6 O’clock position woulld be -90 degrees.

The distance around the circle is called the circumference. and is 2 X radius X Pi.   The symbol for Pi is shown below.   Pi has a value of approximately 3.14

.

The symbol for Pi.

 

Since we have chosen our radius to have a value of 1. the circumference has a value of 2 X Pi * radius or aproximately 6.28

The next thing we do is introduce another way of measuring angles.  This is called radians and is the length of the arc around the circle in multiples of the radius.

 

So this means that the complete trip around the circle (360 degrees) is 2 X Pi radians.   1/2 way around the circle would be Pi radians and a 90 degree angle is Pi / 2 radians.  It is necessary to learn about radians because many software programs require that angles be entered in radians.  The formula for converting from degrees to radians is:

angle in radians = angle in degrees X Pi / 180.

angle in degrees = angle in radians X .180 / Pi

I then create a triangle by rotating the radius by an angle and extending a vertical line down from the point where the radius intersects the circle down to the zero degree line.  We now have a right triangle as shown in the next picture with the line rotated to a 45 degree angle.

Using the names we used in a past episode, the vertical line forming the triangle is the opposite side, and the horizontal line is called the adjacent side.

Now we can use these values and form two new functions.

Opposite / hypotenuse = sine (angle)

Adjacent / hypotenuse = cosine (angle)

You have just learned a little about trigonometry.

Gary

 

 

 

 

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