The point #(8,15)# is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?
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To determine the exact values of the six trigonometric functions of the angle corresponding to the point (8,15) on the terminal side in standard position, we can use the following steps:

First, calculate the length of the radius (r) using the Pythagorean theorem: ( r = \sqrt{x^2 + y^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 )

Next, determine the values of the trigonometric functions using the coordinates (x, y) and the radius (r):
 ( \sin(\theta) = \frac{y}{r} = \frac{15}{17} )
 ( \cos(\theta) = \frac{x}{r} = \frac{8}{17} )
 ( \tan(\theta) = \frac{y}{x} = \frac{15}{8} )
 ( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{17}{15} )
 ( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{17}{8} )
 ( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{8}{15} )
Therefore, the exact values of the six trigonometric functions of the angle corresponding to the point (8,15) on the terminal side in standard position are:
 ( \sin(\theta) = \frac{15}{17} )
 ( \cos(\theta) = \frac{8}{17} )
 ( \tan(\theta) = \frac{15}{8} )
 ( \csc(\theta) = \frac{17}{15} )
 ( \sec(\theta) = \frac{17}{8} )
 ( \cot(\theta) = \frac{8}{15} )
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