Answer
$sin\theta=-\frac{3}{4}$
$cos\theta=\frac{\sqrt 7}{4}$
$tan\theta=-\frac{3}{ \sqrt 7}=-\frac{3\sqrt 7}{ 7}$
$sec \theta=\frac{4}{\sqrt 7}=\frac{4\sqrt 7}{ 7}$
$cot\theta=-\frac{\sqrt 7}{ 3}$
Work Step by Step
Since $csc\theta=-\frac{4}{3}$, we can label the hypotenuse side as having length 4 and the opposite side as having length 3.
The Pythagorean Theorem gives
adjacent side $=\sqrt {4^{2}-3^{2}}=\sqrt 7$
The other five trigonometric functions are given as follows:
$sin\theta=-\frac{3}{4}$
$cos\theta=\frac{\sqrt 7}{4}$
$tan\theta=-\frac{3}{ \sqrt 7}=-\frac{3\sqrt 7}{ 7}$
$sec \theta=\frac{4}{\sqrt 7}=\frac{4\sqrt 7}{ 7}$
$cot\theta=-\frac{\sqrt 7}{ 3}$
