Question
The acute angles in a right triangle are θ
and ϕ
. If sinϕ=0.25
, what is cosθ
?(1 point)
Responses
0.25
0.25
0.75
0.75
0.65
0.65
0.55
and ϕ
. If sinϕ=0.25
, what is cosθ
?(1 point)
Responses
0.25
0.25
0.75
0.75
0.65
0.65
0.55
Answers
GPT-4o mini
To solve for \(\cos \theta\) in a right triangle where the acute angles are \(\theta\) and \(\phi\), and given that \(\sin \phi = 0.25\), we can use the fact that the two acute angles in a right triangle are complementary. Therefore, we have:
\[
\theta + \phi = 90^\circ
\]
This means:
\[
\theta = 90^\circ - \phi
\]
Using the co-function identity, we know that:
\[
\cos \theta = \sin(90^\circ - \phi) = \sin \phi
\]
Since we're given that \(\sin \phi = 0.25\), we can substitute that value in:
\[
\cos \theta = \sin \phi = 0.25
\]
Therefore, the value of \(\cos \theta\) is:
**0.25**
\[
\theta + \phi = 90^\circ
\]
This means:
\[
\theta = 90^\circ - \phi
\]
Using the co-function identity, we know that:
\[
\cos \theta = \sin(90^\circ - \phi) = \sin \phi
\]
Since we're given that \(\sin \phi = 0.25\), we can substitute that value in:
\[
\cos \theta = \sin \phi = 0.25
\]
Therefore, the value of \(\cos \theta\) is:
**0.25**