Question

Using the Pythagorean Identity, determine cosθ if sinθ=7/9 and π/2<θ<π .

Answer 1

To find \( \cos \theta \) given that \( \sin \theta = \frac{7}{9} \) and \( \frac{\pi}{2} < \theta < \pi \), we can use the Pythagorean Identity, which states:

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

First, we square \( \sin \theta \):

\[
\sin^2 \theta = \left(\frac{7}{9}\right)^2 = \frac{49}{81}.
\]

Next, we use the Pythagorean Identity to find \( \cos^2 \theta \):

\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{49}{81}.
\]

To perform the subtraction, we convert \( 1 \) into a fraction with a denominator of 81:

\[
1 = \frac{81}{81},
\]

so we have:

\[
\cos^2 \theta = \frac{81}{81} - \frac{49}{81} = \frac{32}{81}.
\]

Now, we take the square root of both sides to find \( \cos \theta \):

\[
\cos \theta = \pm \sqrt{\frac{32}{81}} = \pm \frac{\sqrt{32}}{9} = \pm \frac{4\sqrt{2}}{9}.
\]

Since \( \theta \) is in the interval \( \frac{\pi}{2} < \theta < \pi \), this means \( \theta \) is in the second quadrant, where the cosine function is negative. Therefore, we have:

\[
\cos \theta = -\frac{4\sqrt{2}}{9}.
\]

Thus, the value of \( \cos \theta \) is:

\[
\boxed{-\frac{4\sqrt{2}}{9}}.
\]