SOLUTION: Use the given information to find sin(2θ), cos(2θ), tan(2θ) for csc(θ)=(-3)/(2).

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Question 366652: Use the given information to find sin(2θ), cos(2θ), tan(2θ) for csc(θ)=(-3)/(2).
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(Note: Since Algebra.com's software does not make using theta easy, I am going to use "x" instead. So you'll have to replace my x's with theta's to match your problems.)

Since sin is the reciprocal of csc, then if csc(x) = %28-3%29%2F2, then sin(x) = %28-2%29%2F3. With this we can find cos(2x). As you probably know, there are three variations of the formula for cos(2x):
cos(2x) = cos%5E2%28x%29+-+sin%5E2%28x%29+=+2cos%5E2%28x%29+-+1+=+1+-+2sin%5E2%28x%29
Since we have the sin(x), we will use the third variation:
cos(2x) = 1+-+2sin%5E2%28x%29+=+1+-+2%2A%28-2%2F3%29%5E2+=+1+-+2%2A%284%2F9%29+=+1+-+8%2F9+=+1%2F9

For sin(2x) there is only one formula:
sin(2x) = 2*sin(x)*cos(x)
We have sin(x) but we do not know what cos(x) is. We could use cos%5E2%28x%29+=+1-sin%5E2%28x%29:
cos%5E2%28x%29+=+1+-+%28-2%2F3%29%5E2+=+1+-+4%2F9+=+5%2F9
Then we can find the square root of each side:
cos%28x%29+=+sqrt%285%29%2F3 or cos%28x%29+=+-sqrt%285%29%2F3
And now we have a little problem. There are two possible values for cos(x)! (This should not be a surprise. If sin(x) is negative then the angle terminates in either the 3rd or 4th quadrants. But cos(x) is negative in the third quadrant
and it is positive in the 4th quadrant. Is it possible you left out part of the problem? Was there something that could be used to determine if x was in the 3rd or 4th quadrants?) So we will have two possible values for sin(2x):
sin(2x) = 2%2A%28-2%2F3%29%2A%28sqrt%285%29%2F9%29+=+-%284sqrt%285%29%29%2F9
or
sin(2x) = 2%2A%28-2%2F3%29%2A%28-sqrt%285%29%2F9%29+=+%284sqrt%285%29%29%2F9

Since tan(2x) = sin(2x)/cos(2x) and since we have two values for sin(2x) we will also end up with two values for tan(2x):
tan(2x) = %28-%284sqrt%285%29%29%2F9%29%2F%285%2F9%29+=+-+%284sqrt%285%29%29%2F5
or
tan(2x) = %28%284sqrt%285%29%29%2F9%29%2F%285%2F9%29+=+%284sqrt%285%29%29%2F5

If you find that the problem does indicate whether x was in the 3rd or 4th quadrant then...
  • if x is in the 3rd quadrant then use the negative value for cos(x) to find sin(2x)
  • if x is in the 4th quadrant then use the positive value for cos(x) to find sin(2x)

Then, use that sin(2x) to find tan(2x).