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See Quick Instructions and Further Discussion below...
This material explains the definitions of the six trigonometric functions in terms of an (x, y) point located on the terminal side of the input angle. You should be familiar with:
The six trigonometric functions (sine, cosine, tangent, cotangent, secant and cosecant) are usually thought to accept an angle as input and output a pure number. For the purposes of the definitions this angle is to be placed in standard position. We will be concerned with any (x, y) point located on the terminal side of this angle. These definitions are based on such an (x, y) point.
These definitions also use the distance from the origin to the (x, y) point. This distance will be referred to as r (for radius) and can be calculated like this:
The six definitions are:
sin(angle) = y/r
cos(angle) = x/r
tan(angle) = y/x (x not equal to zero)
csc(angle) = r/y (y not equal to zero)
sec(angle) = r/x (x not equal to zero)
cot(angle) = x/y (y not equal to zero)
So, for example, the point (5, 7) is on the terminal side of an angle in standard position which has a measure of about 0.95 radians (about 54 degrees):
The distance from the origin to the point (5, 7) can be calculated this way (approximate result given):
Therefore, the six trigonometric function values for this angle can be calculated as follows (approximate results given):
sin(0.95) = y/r = 7/8.6 = 0.81
cos(0.95) = x/r = 5/8.6 = 0.58
tan(0.95) = y/x = 7/5 = 1.4
csc(0.95) = r/y = 8.6/7 = 1.2
sec(0.95) = r/x = 8.6/5 = 1.7
cot(0.95) = x/y = 5/7 = 0.71
In all of the above calculations approximate results were given. Of course, if you are doing more careful and important work you would use measurements and calculations of higher significance.
The definitions given here for the six trigonometric functions are more powerful than the right triangle definitions given in the introduction to trigonometry section. The right triangle definitions are only good for angles up to pi/2 radians (90 degrees). These trig definitions based upon an (x, y) point on the terminal side of an angle are good for angles of any measurement, positive or negative.