Let θ be an angle in quadrant II such that sinθ= (1/4), how do you find the values of secθ and tanθ?

1 Answer
Sep 4, 2016

Answer:

#sectheta = -(4sqrt(15))/15#

#tantheta = -sqrt(15)/15#

Explanation:

Recall that #sintheta = "opposite"/"hypotenuse"#

Hence, the side opposite #theta# in our question measures #1# unit and the hypotenuse measures #4# units.

Since we're dealing with right triangles, we can find the side adjacent #theta# using pythagorean theorem.

Let the adjacent side be #a#.

#a^2 + 1^2 = 4^2#

#a^2 + 1 = 16#

#a^2 = 15#

#a = sqrt(15)#

Now, let's define secant and tangent.

#sectheta = 1/(costheta) = 1/("adjacent"/"hypotenuse") = "hypotenuse"/"adjacent"#

#tantheta = sintheta/costheta = ("opposite"/"hypotenuse")/("adjacent"/"hypotenuse") = "opposite"/"adjacent"#

Applying these definitions:

#sectheta = 4/sqrt(15) = (4sqrt(15))/15#

#tantheta = 1/sqrt(15) = sqrt(15)/15#

The last thing left to do is to find the signs of these ratios. We know that we're in quadrant #II#, where sine is positive, and all the other ratios are negative. Since secant is related to cosine, it will be negative.

So, our final ratios are:

#sectheta = -(4sqrt(15))/15#

#tantheta = -sqrt(15)/15#

Hopefully this helps!