Find sec theta if sin theta = -4/5 and 270 degrees < theta < 360 degrees.
sin theta = -4/5 , so y=-4, r = 5
then r^2 = x^2 + y^2
25 = x^2 + 16
x = ±3 , but we are in quadrant IV, so x = +3
then cos theta = 3/5
and
sec theta = 5/3
We are talking a 3,4,5 right triangle here in the fourth quadrant.
Draw it
opposite down to the right 4 units
hypotenuse 5
adjacent from origin to x = +3
so
cos Theta = 3/5
sec T = 1/cos T = 5/3
To find the value of sec(theta), we need to determine the cosine of theta.
Given that sin(theta) = -4/5, we can use the Pythagorean identity to find the value of cos(theta):
cos^2(theta) = 1 - sin^2(theta)
cos^2(theta) = 1 - (-4/5)^2
cos^2(theta) = 1 - 16/25
cos^2(theta) = (25 - 16) / 25
cos^2(theta) = 9/25
Since theta is in the fourth quadrant (270 degrees < theta < 360 degrees), the cosine is positive. Thus, cos(theta) = 3/5.
Now, to find sec(theta), we can use the reciprocal relationship between secant and cosine:
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (3/5)
sec(theta) = 5/3
Therefore, sec(theta) = 5/3.
To find sec(theta), we need to first determine the value of cos(theta), as sec(theta) is the reciprocal of cos(theta).
Given that sin(theta) = -4/5, we can use the Pythagorean identity to find the value of cos(theta):
cos(theta) = sqrt(1 - sin^2(theta))
cos(theta) = sqrt(1 - (-4/5)^2)
cos(theta) = sqrt(1 - 16/25)
cos(theta) = sqrt(9/25)
cos(theta) = 3/5
Now that we have the value of cos(theta), we can find sec(theta) by taking its reciprocal:
sec(theta) = 1/cos(theta)
sec(theta) = 1/(3/5)
sec(theta) = 5/3
Therefore, sec(theta) = 5/3 when 270 degrees < theta < 360 degrees and sin(theta) = -4/5.