Finding $\sin \theta$ and $\cos \theta$ while vector is given. If **v** = 2 **i** -3 **j** , then find $\sin \theta$ and $ \cos \theta$? Solution manual says: ![enter image description here]( Can someone expound on what solution manual said, I am having hard time understanding it.

This is the definition of $\sin{t}$ and $\cos{t}$ that you should be thinking of:

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This is on the unit circle, so you want to find a vector $\mathbf{w} = (\cos{t},\sin{t}) = \cos{t}\mathbf{i} + \sin{t}\mathbf{j}$ that points in the same direction as $\mathbf{v}$ (so that it preserves the angle $t$) and has length one (so that it lies on the unit circle). This is the normalized vector

$$ \mathbf{w} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{2\mathbf{i} - 3\mathbf{j}}{\sqrt{2^2 + (-3)^3}} = \frac{2}{\sqrt{13}}\mathbf{i} - \frac{3}{\sqrt{13}}\mathbf{j} $$

Therefore,

$$ \cos{t} = \frac{2}{\sqrt{13}} \\\ \sin{t} = - \frac{3}{\sqrt{13}} $$