### Video Transcript

Find the csc of π given that tan
of π equals 24 over seven and [cos] of π is less than zero.

First, letβs think about our trig
identities, sine, cosine, and tangent. Sine is the opposite side length
over the hypotenuse. Cosine is the adjacent side length
over the hypotenuse. Tangent is the opposite side length
over the adjacent side length. Cosecant π is the inverse of sine,
which means itβs the hypotenuse over the opposite side length. Secant is the inverse of cosine,
which means itβs the hypotenuse over the adjacent side. And cotangent is the inverse of
tangent, making it the adjacent side length over the opposite side length.

Weβre given tan of π equals 24
over seven. Since we know that the tangent is
the opposite over adjacent side length and weβre looking for cosecant, which is the
hypotenuse over the opposite, we can go ahead and take the length of 24 and plug
that in. If we call this our π angle, then
we can write 24 and seven as the opposite and adjacent sides. And then, we can use the
Pythagorean theorem to solve for π». According to the Pythagorean
theorem, the hypotenuse squared is equal to side length π squared plus side length
π squared. For us, weβll have seven squared
plus 24 squared equals our hypotenuse squared. Seven squared is 49. 24 squared is 576. When we add 49 plus 576, we get
625. We recognize 625 as a square
number. When we take the square root of
both sides, we find that the hypotenuse is 25. So we plug that in.

This is where we need to be really
careful. Weβre told that the cosine of this
angle is negative. This is telling us something about
the quadrant that this angle falls in. Cosine is only negative in quadrant
two and three. We also know that in quadrant two,
tangent is negative. But in quadrant three, tangent is
positive. We have a positive tangent and a
negative cosine. And that means our angle will fall
in quadrant three, where sine is negative, cosine is negative, and tangent is
positive. Since cosecant is the inverse of
sine and sine is negative, the cosecant will also be negative in this case. The csc of π is negative 25 over
24 under these conditions.