The Unit Circle (radius = 1; 360 ^o): The Number Line wrapped around a coordinate plane, x,y axes, forms the basis for determining and understanding the Trigonometric Functions

The Number Line ( 0 to + infinity in a counter clockwise direction; and 0 to - infinity in a clockwise direction) can be viewed in its simplist form with only four points (x, y values in the coordinate plane):

• Zero position of the Number Line at, (1, 0);
• 1.5707 or π/2 or 90 ^o at, (0, 1);
• 3.1415 or π or 180 ^o at, (-1, 0);
• 4.712 or 3π/2 or 270 ^o at, (0, -1); and
• 6.283 or 2π or 360 ^o at, (1, 0)

You can use this Interactive Unite Circle to determine various angles.

There are four Quadrants within the x, y coordinate plane:

1st Quadrant: Both x and y values are positive; 2nd Quadrant: x is negative, y is positive; 3rd Quadrant: Both x and y are negative; 4th Quadrant: x is positive and y is negative.

With this information, we can define the Six Trigonometric Functions, where the radius, r = 1; and,

θ = an angle in degrees or radians:

1. Sine: sin θ = y/r = y,
2. Cosine: cos θ = x/r =x,
3. Tangent: tan θ = y/x
4. Cotangent: cot θ = x/y,
5. Secant: sec θ = 1/x,
6. Cosecant: csc θ = 1/y.

For example where the radius, r, assumes a value other that r =1, the value of sin θ = y/r can be calculated as follows:

With these definitions of Sine (the y-value on the Unit Circle) and Cosine (the x-value on the Unit Circle), we can now view the Unit Circle (360 degrees) partitioned into various sections equivalent to angles of various measure, in both degrees and radians.

We can now evaluate any Trignometric Function:

• Sin(30 ^o or π/6) = y = 1/2
• Cos(30 ^o or π/6 ) = x = √3/2
• Sin(45 ^o or π/4) = y = Ö2/2
• Cos(45 ^o or π/4) = x = √2/2
• Sin(60 ^o or π/3) = y = √3/2
• Cos(60 ^o or π/3) = x = 1/2
• Sin(90 ^o or π/2) = y = 0
• Cos(90 ^o or π/2) = x = 1
• Tan(30 ^o or π/6) = y/x = 1/2/√3/2 = √3/3
• Sec(30 ^o or π/6) = 1/x = = 1//√3/2 = 2√3/3