Posted by natash on Sunday, September 14, 2008 at 3:26am.
Think of the various definitions of these functions in terms of the height, base and hypoteneuse of a right-angled triangle. You're told that tan(x)=8/3, which means that the ratio of the height to the side is 8 to 3. So you can work out the hypoteneuse by Pythagoras. That should enable you to work out the other functions. Don't forget that condition about the sine of x being less than 0 though: there's more than one angle in a 360-degree circle with a tangent of 8/3, and you need to make sure you pick the right one.
If tan x is positive and sin x is negative then yx must be in the third quadrant. For tan x = 8/3, the reference angle to the -x axis must be arcsin 8/sqrt(8^2 + 3^2) = 69.44 degrees. That means the angle x = 249.44 degrees. The cosine of that angle is -0.3512. The secant is the reciprocal of cosine. csc is the reciprocal of the sin, or -(sqrt73)/8
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