Find the value and angle for the given $csc$ value

Given $csc(\theta) = \frac{5}{3}$, find the corresponding angle $\theta$.

We know:

$csc(\theta) = \frac{1}{sin(\theta)}$

Given,

$\frac{1}{sin(\theta)} = \frac{5}{3}$

So,

$sin(\theta) = \frac{3}{5}$

To find $\theta$, we take the inverse sine:

$\theta = sin^{-1}(\frac{3}{5})$

Using a calculator, we find:

$\theta \approx 36.87^\circ \, or \, \theta \approx 143.13^\circ$

Given $csc( heta) = 2$, find the corresponding angle $ heta$.

We know:

$csc( heta) = frac{1}{sin( heta)}$

Given,

$frac{1}{sin( heta)} = 2$

So,

$sin( heta) = frac{1}{2}$

To find $ heta$, we take the inverse sine:

$ heta = sin^{-1}(frac{1}{2})$

Using a calculator, the principal value is:

$ heta = 30^circ$

Since $csc( heta)$ is positive in both the first and second quadrants:

$ heta = 30^circ , or , heta = 150^circ$

Given $csc(2 heta) = frac{5}{4}$, find the corresponding angle $2 heta$.

We know:

$csc(2 heta) = frac{1}{sin(2 heta)}$

Given,

$frac{1}{sin(2 heta)} = frac{5}{4}$

So,

$sin(2 heta) = frac{4}{5}$

To find $2 heta$, we take the inverse sine:

$2 heta = sin^{-1}(frac{4}{5})$

Using a calculator, we find:

$2 heta approx 53.13^circ$

So,

$ heta approx 26.565^circ$

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