If cot(x)= -16/15, where x is in quadrant IV, what is the exact value of sin(2x)?
tan(x) = 1 / cot(x) = -15 / 16
sin(2x) = [2 tan(x)] / [ 1 + tan^2(x)]
nice one, but why not be even more direct?
sin(2x) = [2 cot(x)] / [ 1 + cot^2(x)]
amazing, innit? Just multiply top and bottom by cot^2(x)
To find the exact value of sin(2x), we first need to find the value of sin(x). Since x is in quadrant IV and cot(x) is given as -16/15, we can use the Pythagorean identity to find sin(x).
The Pythagorean identity states that sin^2(x) + cos^2(x) = 1.
Since cot(x) = cos(x)/sin(x), we can substitute this into the Pythagorean identity:
(-16/15)^2 + cos^2(x) = 1
To solve for cos(x), we can rearrange the equation:
cos^2(x) = 1 - (-16/15)^2
cos^2(x) = 1 - 256/225
cos^2(x) = 225/225 - 256/225
cos^2(x) = -31/225
Since x is in quadrant IV, where cosine is positive, we take the positive square root of cos^2(x):
cos(x) = √(-31/225)
Now, to find sin(x), we can use the identity sin^2(x) + cos^2(x) = 1:
sin^2(x) + (-31/225) = 1
sin^2(x) = 1 + 31/225
sin^2(x) = 256/225
Taking the square root of both sides:
sin(x) = ±√(256/225)
Since x is in quadrant IV, where sine is negative, we take the negative square root:
sin(x) = -√(256/225)
Now, to find sin(2x), we can use the double angle formula:
sin(2x) = 2sin(x)cos(x)
Substituting the values we found:
sin(2x) = 2(-√(256/225))√(-31/225)
Now it's a matter of simplifying this expression to get the exact value of sin(2x).