Answer to Question #343111 in Trigonometry for Sunshine

Question

If cos(𝜃 ) = 3/5 and tan( 𝜃 ) < 0 , find the exact value of each of the remaining trigonometric functions of 𝜃

Answer 1

$\tan\theta=\frac{\sin\theta}{\cos\theta}<0$ and $\cos\theta=\frac 35>0$ so we can conclude $\sin\theta<0$

From $\sin^2\theta+\cos^2\theta=1$ we can write

$\sin\theta=-\sqrt{1-\cos^2\theta}=-\sqrt{1-9/25}=-4/5$

$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac35:(-\frac45)=-\frac34$

$\cot\theta=\frac{1}{\tan\theta}=-\frac43$

$\sec\theta=\frac{1}{\cos\theta}=\frac53$

$\csc\theta=\frac{1}{\sin\theta}=-\frac54$