trigonometric function

Question

Write the first trigonometric function in terms of the second for 𝜃 in the given quadrant.

sec(𝜃), tan(𝜃); 𝜃 in Quadrant III

sec(𝜃)=

Answer 1

find secx in terms of tanx

the Pythagorean Theorem

x^2 + y^2 = r^2

divide by r^2

(x/r)^2 + (y/r)^2 = 1 x/r= cosx, y/r=sinx

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

(where you might have started, if you're familiar with the most basic trig identity)

divide by cos^2(x) to get

tan^2(x) + 1 = sec^2(x)

sec^2(x) = 1+tan^2(x)

secx = -sqr(1+tan^2(x))

secx has to = the negative square root if the angle is in quadrant III

secants of any angle in quadrant III are always <0