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Since some students don't learn the special angle values for sec, csc and cot, I am going to use the fact that csc and sin are reciprocals of each other to rewrite the equation in terms of sin(x). (If you already know what reference angle has a csc of 2, the following steps are unnecessary.)
Multiply by sin(x):
Divide by 2:
We should recognize that 1/2 (positive or negative) is a special angle value for sin. Without our calculator we should know that a 1/2 for sin indicates a reference angle of . Since the 1/2 is positive and since sin is negative in the 1st and 2nd quadrants we should get the following general solution equations: (for the 1st quadrant) (for the 2nd quadrant)
The second equation simplifies:
Now we try various integer values for n as we search for specific solutions which are in the specified interval.
From :
If n = 0 then
If n = 1 (or larger) then x is too large for the interval
If n = -1 (or smaller) then is too small for the interval
From :
If n = 0 then
If n = 1 (or larger) then x is too large for the interval
If n = -1 (or smaller) then x is too small for the interval
So the only solutions in the specified interval are: and .
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