Determine the value of $ an(θ)$ when $sin(θ) = frac{3}{5}$ and $θ$ is in the first quadrant.

Given that $\sin(θ) = \frac{3}{5}$ and $θ$ is in the first quadrant, we can find $\cos(θ)$ using the Pythagorean identity:

$\sin^2(θ) + \cos^2(θ) = 1$

Plugging in the given value:

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

$\frac{9}{25} + \cos^2(θ) = 1$

$\cos^2(θ) = 1 – \frac{9}{25} = \frac{16}{25}$

Since $θ$ is in the first quadrant, $\cos(θ)$ is positive:

$\cos(θ) = \frac{4}{5}$

Now, we can find $\tan(θ)$:

$\tan(θ) = \frac{\sin(θ)}{\cos(θ)} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}$

Therefore, $\tan(θ) = \frac{3}{4}$.

Given $sin(θ) = frac{3}{5}$:

$cos^2(θ) = 1 – sin^2(θ) = 1 – left(frac{3}{5}
ight)^2 = frac{16}{25}$

$cos(θ) = frac{4}{5}$

Then, $ an(θ) = frac{sin(θ)}{cos(θ)} = frac{3}{4}$.

Using $sin(θ) = frac{3}{5}$:

$cos(θ) = frac{4}{5}$

$ an(θ) = frac{3}{4}$.

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