Find csc theta, given that cot theta =-1/2 and theta is in quadrent IV
cotθ = -1/2 or tanθ = -2/1
sketch the triangle in standard position, so
x^2 + y^2 = r^2
r^2 = 1 + 4 = 5
r = √5
if θ is in quad IV, x = 1, y = -2, r = √5
sinθ = y/r = -2/√5
cscθ = - √5/2
Oh, you want to find the cosecant of theta, huh? Well, let's have some fun with numbers then!
We know that cot theta is equal to -1/2. And since theta is in quadrant IV, we can recall that cot theta is positive in that quadrant. So, we have our starting point.
Now, let's think about it. Cotangent is the reciprocal of the tangent function, right? And the cosecant is the reciprocal of the sine function.
So, if cot theta is -1/2, then that means the tangent of theta is its reciprocal, which is -2. Are you following me so far?
Now, since the tangent is negative, we know that the cosine of theta must be positive in quadrant IV. But we're not interested in the cosine right now, we're after the cosecant.
So, let's connect the dots. We know that the sine of theta is the reciprocal of the cosecant, right? And we know that cosine squared plus sine squared equals 1. So, if the cosine is positive and the tangent is negative, can you guess what the sign of the sine would be?
Drumroll, please... it's negative!
So, with a negative sine, we can use the Pythagorean identity to find the sine of theta. We have the cosine as positive (let's say 1 for simplicity), and then the sine squared plus 1 squared equals 1. Solving for sine, we get -√3/2.
And there you have it! The cosecant of theta is the reciprocal of the sine, so it would be -2/√3 (or you can rationalize the denominator to get -2√3/3). Ta-da!
To find the value of csc(theta), we need to use the relationship between trigonometric functions:
csc(theta) = 1 / sin(theta)
Since cot(theta) = -1/2 and theta is in quadrant IV, we can use the Pythagorean identity to find the value of sin(theta):
cot(theta) = 1/tan(theta) = -1/2
tan(theta) = -2
Since theta is in quadrant IV, both sine and cosine are negative in this quadrant. Therefore, we can use the Pythagorean identity to find the value of sin(theta):
sin^2(theta) + cos^2(theta) = 1
Since sine is negative in quadrant IV, we have:
sin^2(theta) + cos^2(theta) = 1
sin^2(theta) + (-cos(theta))^2 = 1
sin^2(theta) + cos^2(theta) = 1
sin^2(theta) = 1 - cos^2(theta)
Since cot(theta) = -1/2, we also have:
cos(theta) / sin(theta) = -1/2
From this equation, we can substitute cos(theta) with -1/2 * sin(theta) :
(-1/2 * sin(theta)) / sin(theta) = -1/2
-1/2 = -1/2
Now, substitute the value of sin^2(theta) from earlier into the equation:
sin^2(theta) = 1 - cos^2(theta)
1 - cos^2(theta) = 1 - (-1/2)^2
1 - cos^2(theta) = 1 - 1/4
cos^2(theta) = 3/4
Taking the square root of both sides, we get:
cos(theta) = ± sqrt(3/4)
cos(theta) = ± sqrt(3)/2
Since theta is in quadrant IV, cosine is positive, so we have:
cos(theta) = sqrt(3)/2
Now, we can find the value of sine using the Pythagorean identity:
sin^2(theta) = 1 - cos^2(theta)
sin^2(theta) = 1 - (sqrt(3)/2)^2
sin^2(theta) = 1 - 3/4
sin^2(theta) = 1/4
Taking the square root of both sides, we get:
sin(theta) = ± 1/2
Since theta is in quadrant IV, sine is negative, so we have:
sin(theta) = -1/2
Finally, substitute the value of sin(theta) into the equation for csc(theta):
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (-1/2)
csc(theta) = -2
Therefore, csc(theta) = -2.
To find the cosecant (csc) of theta, we first need to determine the sine (sin) of theta. Then we can use the reciprocal relationship between sine and cosecant to find csc theta.
Since cot(theta) = -1/2, we can find the tangent (tan) of theta by taking the reciprocal of the cotangent. The reciprocal of -1/2 is -2/1 or -2. Therefore, tan(theta) = -2.
In the fourth quadrant, the cosine (cos) is positive and the sine is negative. Therefore, sin(theta) = -sqrt(1 - cos^2(theta)). To find cos(theta), we can use the Pythagorean identity: cos^2(theta) + sin^2(theta) = 1.
Since cosine is positive in the fourth quadrant, we can let cos(theta) = x, where x is a positive number. Then, cos^2(theta) = x^2, and sin^2(theta) = 1 - x^2.
Using the Pythagorean identity, we have:
x^2 + (1 - x^2) = 1
x^2 + 1 - x^2 = 1
1 = 1
This means that any positive number x can be a solution for cos(theta), as long as the resulting sin(theta) is negative.
Now, let's solve for sin(theta):
sin^2(theta) = 1 - cos^2(theta)
sin^2(theta) = 1 - x^2
sin(theta) = -sqrt(1 - x^2)
Since sin(theta) is negative in the fourth quadrant, we choose the negative square root.
Now, we can find the cosecant (csc) of theta by taking the reciprocal of sin(theta):
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (-sqrt(1 - x^2))
Therefore, csc(theta) = -1 / sqrt(1 - x^2), where x is any positive number.
Since we don't have a specific value for cos(theta), we can express csc(theta) as -1 / sqrt(1 - x^2) in terms of x.