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Study Notes 028:

Trigonometry 2:

By William E. Steinman:

June 4, 2007:

This is a bit of a different look at some of the concepts of plane geometry.

Lt’s look at a plan angle.

The resultant angle is called positive if the rotation is counterclockwise and negative if the rotation is clockwise.

A degree is defined as the measure of the angle of a rotation of 1/360 of the circumference of a circle.

A minute (′) of arc is defined as 1/60 of a degree.

A second (″ ) of arc is defined as 1/60 of a minute.

The circumference of a circle is equal to 2πr where r is the radius of the circle.

If we take an approximate value of π as 3.14159 we get the following.

1 radian (rad) = 180⁰/π = 57.296⁰ = 57⁰17′ 45″ .

1 degree = π/180 rad = 0.017453 rad.

Let us look at a circle.

On this circle of radius r, a central angle of Ө radians intercepts an arc of length x.

The arc length is equal to the product of the radius and the central angle in radians.

x = πӨ

The Unit Circle is a tool used in understanding sins and cosines of angles found in right triangles. It is so named because its radius is exactly one unit in length, usually just called "one". The circle's center is at the origin, and its circumference comprises the set of all points that are exactly one unit from the origin while lying in the plane.

Let’s look at a unit circle.

If (x, y) is a point on the unit circle in the first quadrant, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation X² + Y² = 1.

1² + 1² = 1

Since x² = (−x)² for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not just those in the first quadrant.

Since the radius of a unit circle is 1 the circumference 2πr becomes 2π.

This means the distance halfway around the circle is π and ¼ of the distance is π/2.

Now the arc (S) = rӨ = 1 x Ө = Ө.

Cool.

This means every radian measurement around the circle is paired with a real number point where the points are 1 radius apart in a positive or negative direction.

That is enough background. In SN 029 we can ease into the trigonometric functions of a general angle.

I’ll bet you can hardly wait.