Find the value of $ an( heta)$ given that $sin( heta) = frac{3}{5}$ and $ heta$ is in the second quadrant. Show all steps.

Given that $\sin(\theta) = \frac{3}{5}$, we know the following:

Since $\sin^2(\theta) + \cos^2(\theta) = 1$

$\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1$

$\frac{9}{25} + \cos^2(\theta) = 1$

$\cos^2(\theta) = 1 – \frac{9}{25}$

$\cos^2(\theta) = \frac{16}{25}$

Since $\theta$ is in the second quadrant, $\cos(\theta)$ is negative:

$\cos(\theta) = -\frac{4}{5}$

Now, $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

$\tan(\theta) = \frac{\frac{3}{5}}{-\frac{4}{5}}$

$\tan(\theta) = -\frac{3}{4}$

So, $\tan(\theta) = -\frac{3}{4}$.

We start with $sin( heta) = frac{3}{5}$ and find $cos( heta)$ using the Pythagorean identity:

$sin^2( heta) + cos^2( heta) = 1$

$left(frac{3}{5}
ight)^2 + cos^2( heta) = 1$

$frac{9}{25} + cos^2( heta) = 1$

$cos^2( heta) = frac{16}{25}$

In the second quadrant, $cos( heta)$ is negative:

$cos( heta) = -frac{4}{5}$

Next, calculate $ an( heta)$:

$ an( heta) = frac{sin( heta)}{cos( heta)}$

$ an( heta) = frac{frac{3}{5}}{-frac{4}{5}}$

$ an( heta) = -frac{3}{4}$

The value of $ an( heta)$ is $-frac{3}{4}$.

Given $sin( heta) = frac{3}{5}$, find $cos( heta)$:

$sin^2( heta) + cos^2( heta) = 1$

$cos^2( heta) = 1 – left(frac{3}{5}
ight)^2$

$cos^2( heta) = frac{16}{25}$

Since $ heta$ is in the second quadrant, $cos( heta) = -frac{4}{5}$

Calculate $ an( heta)$:

$ an( heta) = frac{sin( heta)}{cos( heta)} = frac{3/5}{-4/5} = -frac{3}{4}$

Thus, $ an( heta) = -frac{3}{4}$.

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