document.write( "Question 636737: If the point P (-3, 4) is on the terminal sideof theta, find the values of the six trigonometric functions at the angle theta. Give exact answers. \n" ); document.write( "

Algebra.Com's Answer #401194 by jsmallt9(3758) $\"\"$ $\"About$

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The Trig ratios are usually taught in terms of opposite, adjacent and hypotenuse. They can also be taught in terms of x and y coordinates and distance from the origin. The correspondence of these is:
\n" ); document.write( "x-coordinate = adjacent
\n" ); document.write( "y-coordinate = opposite
\n" ); document.write( "distance from the origin = hypotenuse

\n" ); document.write( "The distance from the origin to some point (x, y) can be expressed using the distance formula. This particular distance is usually called \"r\" (for reasons you might be able to figure out.). So
\n" ); document.write( " $\"r+=+sqrt%28%28x-0%29%5E2%2B%28y-0%29%5E2%29+=+sqrt%28x%5E2%2By%5E2%29\"$
\n" ); document.write( "Note: Although x and y coordinates may be negative, r is a distance and it is never negative.

\n" ); document.write( "Using the x and y coordiantes given to us in this problem, we can calculate our value for r:
\n" ); document.write( " $\"r+=+sqrt%28%28-3%29%5E2%2B%284%29%5E2%29+=+sqrt%289%2B16%29+=+sqrt%2825%29+=+5\"$

\n" ); document.write( "We are now ready to find the values of the trig functions. For each function we will replace \"adjacent\" with the x-coordinate, \"opposite\" with the y-coordinate, and \"hypotenuse\" with r we get:
\n" ); document.write( " $\"sin%28theta%29+=+y%2Fr+=+4%2F5\"$
\n" ); document.write( " $\"cos%28theta%29+=+x%2Fr+=+%28-3%29%2F5+=+-3%2F5\"$
\n" ); document.write( " $\"tan%28theta%29+=+y%2Fx+=+4%2F%28-3%29+=+-4%2F3\"$
\n" ); document.write( " $\"csc%28theta%29+=+r%2Fy+=+5%2F4\"$
\n" ); document.write( " $\"sec%28theta%29+=+r%2Fx+=+5%2F%28-3%29+=+-5%2F3\"$
\n" ); document.write( " $\"cot%28theta%29+=+x%2Fy+=+%28-3%29%2F4+=+-3%2F4\"$ \n" ); document.write( "

\n" );