# How do you solve (5cscx)/3 = 9/4 for 0 < x < 2pi rounded to the nearest hundredth of a radian ?

Apr 10, 2018

$x = 0.83 , 2.31$

#### Explanation:

We have: $\frac{5 \csc \left(x\right)}{3} = \frac{9}{4}$; $0 < x < 2 \pi$

$R i g h t a r r o w 5 \csc \left(x\right) = \frac{27}{4}$

$R i g h t a r r o w \csc \left(x\right) = \frac{27}{20}$

$\csc \left(x\right)$ is the reciprocal of $\sin \left(x\right)$, namely $\csc \left(x\right) = \frac{1}{\sin \left(x\right)}$:

$R i g h t a r r o w \frac{1}{\sin \left(x\right)} = \frac{27}{20}$

$R i g h t a r r o w \sin \left(x\right) = \frac{20}{27}$

Let the reference angle be $x = \arcsin \left(\frac{20}{27}\right) = 0.834172325$.

Then, the value of $\sin \left(x\right)$ is $\frac{20}{27}$, which is a positive value.

So, the angles $x$ are located in the first and second quadrants:

$R i g h t a r r o w x = 0.834172325 , \pi - 0.834172325$

$R i g h t a r r o w x = 0.834172325 , 2.307420329$

$\therefore x \approx 0.83 , 2.31$

Therefore, the solutions to the equation, rounded to the nearest hundredth of a radian, are $x = 0.83$ and $x = 2.31$.