Sin theta = 1/4, what is sec theta?, tan theta?

give exact answers for
0 less than or equal to theta less than equal to pi/2

sinØ = 1/4, so opposite = 1, hypotenuse = 4

by Pythagoras, adjacent = √15

secØ = 1/cosØ = 1/(√15/4) = 4/√15
tan Ø = 1/√15

To find the values of sec(theta) and tan(theta), we need to first determine the value of cos(theta), as sec(theta) = 1/cos(theta) and tan(theta) = sin(theta)/cos(theta).

Given that sin(theta) = 1/4, we can solve for cos(theta) using the Pythagorean Identity:

cos^2(theta) + sin^2(theta) = 1

cos^2(theta) + (1/4)^2 = 1
cos^2(theta) + 1/16 = 1
cos^2(theta) = 1 - 1/16
cos^2(theta) = 15/16

cos(theta) = ±√(15/16)
Since theta is in the first quadrant (0 ≤ theta ≤ π/2), the cosine function is positive, so we can take the positive root:

cos(theta) = √(15/16) = √15/4

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

sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (√15/4)
sec(theta) = 4 / √15

tan(theta) = sin(theta) / cos(theta)
tan(theta) = (1/4) / (√15/4)
tan(theta) = 1 / √15

So, the exact values of sec(theta) and tan(theta) when 0 ≤ theta ≤ π/2 are:
sec(theta) = 4 / √15
tan(theta) = 1 / √15

To find the values of sec(theta) and tan(theta) given sin(theta) = 1/4, we can use the trigonometric identities.

First, let's solve for cos(theta) using the identity cos^2(theta) + sin^2(theta) = 1. Substitute sin(theta) = 1/4.

cos^2(theta) + (1/4)^2 = 1
cos^2(theta) + 1/16 = 1
cos^2(theta) = 1 - 1/16
cos^2(theta) = 15/16
Taking the square root of both sides, we get:
cos(theta) = ± sqrt(15)/4

Since 0 ≤ theta ≤ pi/2, we know that theta is in the first quadrant, where cos(theta) is positive. Therefore, we choose the positive value for cos(theta):

cos(theta) = sqrt(15)/4

Now, knowing sin(theta) = 1/4 and cos(theta) = sqrt(15)/4, we can use the identities sec(theta) = 1/cos(theta) and tan(theta) = sin(theta)/cos(theta) to find the values:

sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (sqrt(15)/4)
sec(theta) = 4 / sqrt(15)
Rationalizing the denominator, we get:
sec(theta) = (4 / sqrt(15)) * (sqrt(15) / sqrt(15))
sec(theta) = 4sqrt(15) / 15

tan(theta) = sin(theta) / cos(theta)
tan(theta) = (1/4) / (sqrt(15)/4)
tan(theta) = 1 / sqrt(15)
Rationalizing the denominator, we get:
tan(theta) = (1 / sqrt(15)) * (sqrt(15) / sqrt(15))
tan(theta) = sqrt(15) / 15

Therefore, for 0 ≤ theta ≤ pi/2, sec(theta) = 4sqrt(15) / 15 and tan(theta) = sqrt(15) / 15.