I'm given the problem:
If $\cot(\theta) = 1.5$ and $\theta$ is in quadrant 3, what is the value of $\sin(\theta)$?
I looked at all the related answers I could find on here, but I haven't been able to piece together the answer I need from them.
I know that $\sin^2\theta + \cos^2\theta = 1$, $ \cot^2\theta + 1 = \csc^2\theta $, and $\csc^2\theta = \frac{1}{\sin^2\theta}$
Substituting 3.25 for $\cot^2\theta + 1$ and $\frac{1}{\sin^2\theta}$ for $\csc^2\theta$ I get:
$3.25 = \frac{1}{\sin^2\theta}$
then
$\sin\theta = -\sqrt{\frac{1}{3.25}}$
This doesn't seem correct though. Can anyone help please?
edit: Sorry, meant to make that answer negative.
Best Answer
Indeed, we know that $$1.5 = \cot\theta = \frac{\cos\theta}{\sin\theta}$$ hence $$1.5\sin\theta = \cos\theta.$$ Squaring both sides we have $$2.25\sin^2\theta = \cos^2\theta$$ and since $\cos^2\theta = 1-\sin^2\theta$ we have $$\begin{align*} 2.25\sin^2\theta &= 1-\sin^2\theta\\ 2.25\sin^2\theta + \sin^2\theta &= 1\\ 3.25\sin^2\theta &= 1. \end{align*}$$ From this, you can figure out the value of $\sin^2\theta$. Taking square roots will tell you something about the absolute value of $\sin\theta$.
Now... why did they tell you $\theta$ was in the third quadrant?