Video: Using the Pythagorean Identities to Evaluate a Trigonometric Function given a Trigonometric Function and Quadrant of an Angle

Video Transcript

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

First, let’s 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 𝜃 is the inverse of sine, which means it’s the hypotenuse over the opposite side length. Secant is the inverse of cosine, which means it’s the hypotenuse over the adjacent side. And cotangent is the inverse of tangent, making it the adjacent side length over the opposite side length.

We’re given tan of 𝜃 equals 24 over seven. Since we know that the tangent is the opposite over adjacent side length and we’re 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 𝜃 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 𝐻. According to the Pythagorean theorem, the hypotenuse squared is equal to side length 𝑎 squared plus side length 𝑏 squared. For us, we’ll 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. We’re 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 𝜃 is negative 25 over 24 under these conditions.