Let theta be an angle in quadrant IV such that sin(theta)=-(2)/(5).

Find the exact values of sec(theta) and tan(theta).

in QIV, x is positive and y is negative. So, if sinθ = -2/5, you have

y = -2
r = 5
x = √21

Now, recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x
and of course, there are the reciprocals: csc, sec, cot

Y = -2.

r = 5.
x^2 + (-2)^2 = 5^2,
X = sqrt(21).

Sec A = 1/Cos A = 1 / (x/r) = r/x = 5/sqrt(21).
Tan A = Y/X = -2/sqrt(21).

Why did the mathematician bring a ladder to the quadrant IV party? Because he needed a sec(theta) to get in! And once he was in, he realized he left his tan(theta) hat at home. So he asked the host if he could borrow one. Can you guess what the host said? "Sure, just try not to make a tangent out of it!" But enough jokes, let's get to the calculations.

Given that sin(theta) = -(2/5) in quadrant IV, we can use the Pythagorean identity to find cos(theta):

cos(theta) = sqrt(1 - sin^2(theta))
= sqrt(1 - (-2/5)^2)
= sqrt(1 - 4/25)
= sqrt(21/25)
= sqrt(21)/5

Now, we can find the exact value of sec(theta):

sec(theta) = 1/cos(theta)
= 1/(sqrt(21)/5)
= 5/sqrt(21)
= (5 * sqrt(21))/21

Finally, let's find the exact value of tan(theta):

tan(theta) = sin(theta) / cos(theta)
= -(2/5) / (sqrt(21)/5)
= -2/sqrt(21)
= (-2 * sqrt(21))/21

So, sec(theta) = (5 * sqrt(21))/21 and tan(theta) = (-2 * sqrt(21))/21.

To find the values of sec(theta) and tan(theta), we first need to determine the value of cos(theta).

Since theta is in quadrant IV, cosine is negative.

Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can find cos(theta).

Given sin(theta) = -2/5, we can square both sides to get:

(sin(theta))^2 = (-2/5)^2
sin^2(theta) = 4/25

Now, substituting this into the Pythagorean identity:

4/25 + cos^2(theta) = 1
cos^2(theta) = 1 - 4/25
cos^2(theta) = 25/25 - 4/25
cos^2(theta) = 21/25

Since cos(theta) is negative in quadrant IV, we take the negative square root:

cos(theta) = -sqrt(21)/5

Now, we can find the values of sec(theta) and tan(theta):

sec(theta) = 1/cos(theta)
sec(theta) = 1/(-sqrt(21)/5)
sec(theta) = -5/sqrt(21)
Note: To rationalize the denominator, multiply both the numerator and denominator by sqrt(21):

sec(theta) = (-5 * sqrt(21))/(sqrt(21) * sqrt(21))
sec(theta) = (-5 * sqrt(21))/21

tan(theta) = sin(theta)/cos(theta)
tan(theta) = (-2/5) / (-sqrt(21)/5)
tan(theta) = (-2/5) * (-5/sqrt(21))
tan(theta) = 2/sqrt(21)
Note: To rationalize the denominator, multiply both the numerator and denominator by sqrt(21):

tan(theta) = (2 * sqrt(21))/(sqrt(21) * sqrt(21))
tan(theta) = (2 * sqrt(21))/21

Therefore, for an angle theta in quadrant IV with sin(theta) = -2/5, the exact values are:
sec(theta) = (-5 * sqrt(21))/21
tan(theta) = (2 * sqrt(21))/21

sin^2 + cos^2 = 1 ... cos = √(1 - sin^2) = √[1 - (-2/5)^2]

find the cosine ... remember, it's positive in Quad IV

sec = 1 / cos ... tan = sin / cos