let theta be an angle of quadrant IV such that sin theta = -8/9 find the exact value of sec theta and cot theta?
third leg along x axis is sqrt (81-64) =sqrt (17)
so
x = sqrt (17)
y = -8
hypotenuse = 9
cos = sqrt (17) / 9
so sec = 1/cos = 9/sqrt 17
cot = sqrt (17) / -8
Oh, it seems we have a funny situation here! In quadrant IV, the sine of an angle is negative, but it seems like we have a naughty value for sine here. The sine of any angle, no matter which quadrant it's in, will always be between -1 and 1. So, unfortunately, we can't find the exact value of sec(theta) or cot(theta) with sin(theta) = -8/9. It looks like someone tried to sneak in a little math trick on us! Keep those math jokes coming!
To find the exact values of sec(theta) and cot(theta), we first need to determine the values of cosine(theta) and tangent(theta).
Given that sin(theta) = -8/9 and theta is an angle in quadrant IV, we can use the Pythagorean identity to find the value of cosine(theta):
sin^2(theta) + cos^2(theta) = 1
(-8/9)^2 + cos^2(theta) = 1
64/81 + cos^2(theta) = 1
cos^2(theta) = 17/81
Since theta is in quadrant IV, cosine(theta) is positive. Taking the square root and considering the positive value, we have:
cos(theta) = sqrt(17/81)
cos(theta) = sqrt(17)/9
Now, we can find the value of tangent(theta):
tan(theta) = sin(theta) / cos(theta)
tan(theta) = (-8/9) / (sqrt(17)/9)
tan(theta) = -8 / sqrt(17)
To find sec(theta), we can use the reciprocal relationship between secant and cosine:
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (sqrt(17)/9)
sec(theta) = 9 / sqrt(17)
Finally, we can find cot(theta) using the reciprocal relationship between cotangent and tangent:
cot(theta) = 1 / tan(theta)
cot(theta) = 1 / (-8 / sqrt(17))
cot(theta) = -sqrt(17) / 8
Hence, the exact value of sec(theta) is 9/sqrt(17), and the exact value of cot(theta) is -sqrt(17)/8.
To find the exact values of sec(theta) and cot(theta), we first need to find the value of cos(theta) and tan(theta).
We are given that sin(theta) = -8/9. Since theta is an angle in the fourth quadrant, sine is negative in this quadrant. Therefore, we can determine that cos(theta) will be positive.
Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can solve for cos(theta):
(-8/9)^2 + cos^2(theta) = 1
64/81 + cos^2(theta) = 1
cos^2(theta) = 81/81 - 64/81
cos^2(theta) = 17/81
cos(theta) = √(17/81)
cos(theta) = √17/9
Now that we know cos(theta), we can find the value of sec(theta) using the identity sec(theta) = 1/cos(theta):
sec(theta) = 1/(√17/9)
sec(theta) = 9/√17
To rationalize the denominator, we multiply the numerator and denominator by √17:
sec(theta) = 9√17/9
sec(theta) = √17
Next, we need to find the value of tan(theta). Since tan(theta) = sin(theta)/cos(theta), we can substitute the given values:
tan(theta) = (-8/9)/(√17/9)
tan(theta) = -8/√17
To rationalize the denominator, we multiply the numerator and denominator by √17:
tan(theta) = -8√17/17
Finally, we can use the identity cot(theta) = 1/tan(theta) to find the value of cot(theta):
cot(theta) = 1/(-8√17/17)
cot(theta) = -17/8√17
To rationalize the denominator, we multiply the numerator and denominator by √17:
cot(theta) = -17√17/8
Therefore, the exact values of sec(theta) and cot(theta) are √17 and -17√17/8, respectively.
sin(Θ) = -8/9 = y/r
x = √(r^2 - y^2) = √17
sec(Θ) = r / x
cot(Θ) = x / y