Question
What is the sine for an angle that has a cosine of 25 and is in Quadrant IV? Use the Pythagorean identity sin2θ+cos2θ=1 and the quadrant to solve.(1 point)
Responses
−2125
negative Start Fraction 21 over 25 End Fraction
2125
Start Fraction 21 over 25 End Fraction
21√5
Start Fraction Start Root 21 End Root over 5 End Fraction
−21√5
Responses
−2125
negative Start Fraction 21 over 25 End Fraction
2125
Start Fraction 21 over 25 End Fraction
21√5
Start Fraction Start Root 21 End Root over 5 End Fraction
−21√5
Answers
GPT-4o mini
To find the sine of an angle in Quadrant IV with a given cosine value, we can use the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given that \(\cos \theta = 25\), we first need to recognize that this must actually mean \(\cos \theta = \frac{25}{25} = 1\) or it's likely stated incorrectly. However, assuming we work with \(\cos^2 \theta = \left(\frac{25}{25}\right)^2 = 1\):
Now, let's proceed with calculating \(\sin \theta\) given that \(\cos \theta = 0.25\) (as cosine cannot directly equal 25):
1. Calculate \(\sin^2 \theta\):
\[
\sin^2 \theta = 1 - \cos^2 \theta
\]
\[
\cos^2 \theta = \left(\frac{25}{25}\right)^2 = \frac{625}{625}
\]
Putting the cosine value into the identity:
2. Going through the calculation:
\[
\sin^2 \theta = 1 - \left(\frac{25}{25}\right)^2
\]
Since cosine (if interpreted correctly) would usually involve component squares, we assume a fraction (like \(\frac{5}{25}\)), or here:
\[
\sin^2 \theta = 1 - 0.25^2
\]
\[
= 1 - 0.0625 = 0.9375
\]
\[
= \frac{15}{25}
\]
Resulting in:
\[
\sin \theta = -\sqrt{\frac{21}{25}} \text{ or } -\left(\frac{\sqrt{21}}{5}\right)
\]
Therefore, since sine is negative in Quadrant IV:
\[
\sin \theta = -\frac{\sqrt{21}}{5}
\]
Thus the correct choice that matches the provided options is:
\[
\boxed{-\frac{21\sqrt{5}}{5}}
\]
or simply negative \(\frac{21}{25}\) leading closest here as \( \frac{21}{25} \) is the value we need ultimately consolidating to values reflecting simple trigonomic oversights.
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given that \(\cos \theta = 25\), we first need to recognize that this must actually mean \(\cos \theta = \frac{25}{25} = 1\) or it's likely stated incorrectly. However, assuming we work with \(\cos^2 \theta = \left(\frac{25}{25}\right)^2 = 1\):
Now, let's proceed with calculating \(\sin \theta\) given that \(\cos \theta = 0.25\) (as cosine cannot directly equal 25):
1. Calculate \(\sin^2 \theta\):
\[
\sin^2 \theta = 1 - \cos^2 \theta
\]
\[
\cos^2 \theta = \left(\frac{25}{25}\right)^2 = \frac{625}{625}
\]
Putting the cosine value into the identity:
2. Going through the calculation:
\[
\sin^2 \theta = 1 - \left(\frac{25}{25}\right)^2
\]
Since cosine (if interpreted correctly) would usually involve component squares, we assume a fraction (like \(\frac{5}{25}\)), or here:
\[
\sin^2 \theta = 1 - 0.25^2
\]
\[
= 1 - 0.0625 = 0.9375
\]
\[
= \frac{15}{25}
\]
Resulting in:
\[
\sin \theta = -\sqrt{\frac{21}{25}} \text{ or } -\left(\frac{\sqrt{21}}{5}\right)
\]
Therefore, since sine is negative in Quadrant IV:
\[
\sin \theta = -\frac{\sqrt{21}}{5}
\]
Thus the correct choice that matches the provided options is:
\[
\boxed{-\frac{21\sqrt{5}}{5}}
\]
or simply negative \(\frac{21}{25}\) leading closest here as \( \frac{21}{25} \) is the value we need ultimately consolidating to values reflecting simple trigonomic oversights.