Answer to Question #176084 in Trigonometry for Nash

Question

If cos(𝜃 ) = 1/3 and tan( 𝜃 ) < 0 , find the exact value of each of the remaining trigonometric functions of 𝜃

Answer 1

if cos $\theta$ >0 and tan $\theta$ <0 then this falls on the fourth quadrant. The sin will also be negative.

let x be the adjacent, y the opposite and r the hypotenuse of a right angled triangle where $\theta$ is the angle formed as x and r intersect.

$cos \theta=\frac x r$

x=1, r=3

$y=\sqrt {r^2-x^2}$

$y=\sqrt{3^2-1^2}=\sqrt 8$

$sec \theta=\frac{1}{cos \theta}=\frac{1}{\frac 1 3}=3$

$\sin \theta =-\frac y r=- \frac {\sqrt{8}}3$

$\csc \theta =\frac 1 {sin\theta}=\frac {1}{-\frac {\sqrt 8}{3}}=-\frac{3}{\sqrt 8}$

$\tan \theta =-\frac y x=-\frac {\sqrt 8} 1=-\sqrt 8$

$\cot \theta=\frac{1}{tan \theta}=-\frac{1}{\sqrt 8}$