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

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Answer 1 · 2018-04-10T21:32:27

$x = 0.83, 2.31$

We have: $frac(5 csc(x))(3) = frac(9)(4)$ ; $0 < x < 2 pi$

$Rightarrow 5 csc(x) = frac(27)(4)$

$Rightarrow csc(x) = frac(27)(20)$

$csc(x)$ is the reciprocal of $sin(x)$ , namely $csc(x) = frac(1)(sin(x))$ :

$Rightarrow frac(1)(sin(x)) = frac(27)(20)$

$Rightarrow sin(x) = frac(20)(27)$

Let the reference angle be $x = arcsin(frac(20)(27)) = 0.834172325$ .

Then, the value of $sin(x)$ is $frac(20)(27)$ , which is a positive value.

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

$Rightarrow x = 0.834172325, pi - 0.834172325$

$Rightarrow x = 0.834172325, 2.307420329$

$therefore x approx 0.83, 2.31$

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