If
cot(x) = −19/18
,
where x is in Quadrant IV, what is the exact value of sin(2x)?
tan x = -18/19
hypotenuse = sqrt (18^2+19^2)
= sqrt (324+361)
= sqrt(685)
sin x = -18/sqrt(685)
cos x = +19/sqrt(685)
sin (2x) = 2 sin x cos x
= -2*18*19/685
To find the exact value of sin(2x), we need to use the trigonometric identity for sin(2x) in terms of cot(x). The identity is:
sin(2x) = 2 * sin(x) * cos(x)
Given that cot(x) = -19/18, we can use the relationship between cotangent and sine to find sin(x). The relationship is:
cot(x) = cos(x) / sin(x)
Rearranging the equation, we have:
cos(x) = cot(x) * sin(x)
Now substitute the given value of cot(x) into the equation:
cos(x) = (-19/18) * sin(x)
Since x is in Quadrant IV, where both sine and cosine are negative, we can take the negative value of both sides:
-cos(x) = (19/18) * sin(x)
Now, let's use the Pythagorean identity to find the value of cos(x) in terms of sin(x) in Quadrant IV:
cos^2(x) + sin^2(x) = 1
Since sin(x) is negative in Quadrant IV, we have:
cos^2(x) + (-sin(x))^2 = 1
cos^2(x) + sin^2(x) = 1
Substituting (-cos(x)) for cos(x) in the equation, we get:
(-cos(x))^2 + sin^2(x) = 1
cos^2(x) + sin^2(x) = 1
We notice that this equation is the same as the Pythagorean identity, which means that the squared values of sine and cosine add up to 1. Therefore, the equation holds for any angle x, including x in Quadrant IV.
Now, let's substitute the expression for cos(x) we obtained earlier into the Pythagorean identity:
(-cos(x))^2 + sin^2(x) = 1
((-19/18) * sin(x))^2 + sin^2(x) = 1
Expanding and simplifying the equation:
361/324 * sin^2(x) + sin^2(x) = 1
(361/324 + 1) * sin^2(x) = 1
(685/324) * sin^2(x) = 1
To solve for sin(x), divide both sides by (685/324):
sin^2(x) = 324/685
Taking the positive square root of both sides:
sin(x) = √(324/685)
Now that we have the value of sin(x), we can substitute it back into the formula for sin(2x):
sin(2x) = 2 * sin(x) * cos(x)
Since we already know that cos(x) = (-19/18) * sin(x), we can substitute the values:
sin(2x) = 2 * (√(324/685)) * ((-19/18) * √(324/685))
Simplifying the expression:
sin(2x) = -38√(324/685) / 18
To get the exact value, we can simplify the expression further:
sin(2x) = -19√(9/685)
Therefore, the exact value of sin(2x) when cot(x) = -19/18 and x is in Quadrant IV is -19√(9/685).
To find the exact value of sin(2x), we can use the double-angle identity for sine.
The double-angle identity for sine states that sin(2x) = 2sin(x)cos(x).
Given that cot(x) = -19/18, we can find the values of sin(x) and cos(x) using the relationship between cotangent, sine, and cosine.
cot(x) = cos(x)/sin(x)
From the given information, we have:
-19/18 = cos(x)/sin(x)
To find sin(x) and cos(x), we can use the Pythagorean identity for sine and cosine:
sin^2(x) + cos^2(x) = 1
From the equation sin^2(x) + cos^2(x) = 1, we can solve for cos(x):
cos^2(x) = 1 - sin^2(x)
cos(x) = ±√(1 - sin^2(x))
Since x is in Quadrant IV (where cosine is positive), we have:
cos(x) = √(1 - sin^2(x))
Substituting this value of cos(x) into the cot(x) equation, we get:
-19/18 = (√(1 - sin^2(x))) / sin(x)
Squaring both sides of the equation to eliminate the square root, we get:
361/324 = (1 - sin^2(x)) / sin^2(x)
361sin^2(x) = 324 - 324sin^2(x)
685sin^2(x) = 324
sin^2(x) = 324/685
sin(x) = ±√(324/685) = ±(18/√(5*137))
Since x is in Quadrant IV (where sine is negative), sin(x) = -18/√(5*137).
Using the double-angle identity sin(2x) = 2sin(x)cos(x), we can find the value of sin(2x):
sin(2x) = 2 * (-18/√(5*137)) * √(1 - (-18/√(5*137))^2)
Simplifying the expression gives:
sin(2x) = 2 * (-18/√(5*137)) * √(1 - 324/5*137)
sin(2x) = -36/√(5*137) * √(1 - (324/685))
sin(2x) = -36/√(5*137) * √((685 - 324)/685)
sin(2x) = -36/√(5*137) * √(361/685)
sin(2x) = -36/√(5*137) * √(19/37)
sin(2x) = -36/√(5*137) * (√19 / √37)
sin(2x) = -36√19 / √(5*137*37)
Therefore, the exact value of sin(2x) is -36√19 / √(5*137*37).