Question

Using the Pythagorean Identity, determine cosθ if sinθ=79 and π2<θ<π .(1 point) Responses −42√9 negative Start Fraction 4 Start Root 2 End Root over 9 End Fraction 2√3 Start Fraction Start Root 2 End Root over 3 End Fraction 42√9 Start Fraction 4 Start Root 2 End Root over 9 End Fraction −2√3

Answer 1

To determine \( \cos \theta \) using the Pythagorean Identity, we start with the following identity:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Given that \( \sin \theta = 79 \), we first substitute that into the equation, but it seems there might be an issue here because \( \sin \theta \) must be in the range of \(-1\) to \(1\). Therefore, \( \sin\theta = 79 \) is not a valid value for sine.

Could you please double-check the value of \( \sin \theta \)? If it's indeed supposed to be something like \( \sin \theta = \frac{79}{80} \) or some other value within the valid range, we can proceed from there.

If the value is \( \sin \theta = \frac{79}{80} \), then:

1. Calculate \( \sin^2 \theta \):

\[
\sin^2 \theta = \left(\frac{79}{80}\right)^2 = \frac{6241}{6400}
\]

2. Now use the Pythagorean Identity:

\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{6241}{6400} = \frac{6400 - 6241}{6400} = \frac{159}{6400}
\]

3. Taking the square root to find \( \cos \theta \):

\[
\cos \theta = \pm \sqrt{\frac{159}{6400}} = \pm \frac{\sqrt{159}}{80}
\]

Since \( \theta \) is in the interval \( \left(\frac{\pi}{2}, \pi\right) \), we know \( \cos \theta \) must be negative. Thus:

\[
\cos \theta = -\frac{\sqrt{159}}{80}
\]

Please clarify the value of \( \sin \theta \) if \( 79 \) is not correct, and provide further information if required.