In a right triangle, the sum of the acute angles \( \theta \) and \( \phi \) is \( 90^\circ \). This means \( \theta = 90^\circ - \phi \).
Given \( \sin \phi = 0.25 \), we can use the trigonometric identity for cosine:
\[ \cos \theta = \cos(90^\circ - \phi) = \sin \phi \]
So, we have:
\[ \cos \theta = \sin \phi = 0.25 \]
However, we need to find \( \cos \theta \) in terms of \( \phi \). We can also find \( \cos \phi \) through the Pythagorean identity:
\[ \sin^2 \phi + \cos^2 \phi = 1 \]
Substituting \( \sin \phi \):
\[ (0.25)^2 + \cos^2 \phi = 1 \] \[ 0.0625 + \cos^2 \phi = 1 \] \[ \cos^2 \phi = 1 - 0.0625 = 0.9375 \] \[ \cos \phi = \sqrt{0.9375} \approx 0.968245 \]
Now, we can find \( \cos \theta \):
\[ \cos \theta = \sin \phi = 0.25 \]
So for the response regarding \( \cos \theta \):
You'll find that the answer for \( \cos \theta \) is \( \cos(90^\circ - \phi) = \sin \phi = 0.25 \).
So the correct answer is:
0.25
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