We know that the sine and cosine of an angle \(\theta\) are related by the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
From the problem, we have:
\[ \cos \theta = 25 \]
However, this value needs to be within the range of the cosine function, which is -1 to 1. Thus, it seems there might have been a misunderstanding or misstatement of the value. In a typical context, let's suppose you meant \(\cos \theta = \frac{3}{5}\) instead, which fits within the cosine range and is plausible for Quadrant IV.
Using the Pythagorean identity:
\[ \sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1 \] \[ \sin^2 \theta + \frac{9}{25} = 1 \] \[ \sin^2 \theta = 1 - \frac{9}{25} \] \[ \sin^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \] \[ \sin \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \]
Since we are in Quadrant IV, where the sine is negative, we have:
\[ \sin \theta = -\frac{4}{5} \]
If you were indeed referring to a different cosine value or something else entirely, please clarify the cosine value accurately.
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