Question

if sin0=1/6 and pie/2 <0<pie find cos0

Answer 1

To find the value of \(\cos \theta\) given that \(\sin \theta = \frac{1}{6}\) and \(\frac{\pi}{2} < \theta < \pi\), we can use the Pythagorean identity:

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

First, we can find \(\sin^2 \theta\):

\[
\sin^2 \theta = \left(\frac{1}{6}\right)^2 = \frac{1}{36}
\]

Next, substitute \(\sin^2 \theta\) into the Pythagorean identity:

\[
\frac{1}{36} + \cos^2 \theta = 1
\]

Now, subtract \(\frac{1}{36}\) from both sides:

\[
\cos^2 \theta = 1 - \frac{1}{36}
\]

To combine the fractions, we find a common denominator (which is 36):

\[
1 = \frac{36}{36}
\]
\[
\cos^2 \theta = \frac{36}{36} - \frac{1}{36} = \frac{35}{36}
\]

Now, take the square root of both sides:

\[
\cos \theta = \pm \sqrt{\frac{35}{36}} = \pm \frac{\sqrt{35}}{6}
\]

Since \(\frac{\pi}{2} < \theta < \pi\), \(\theta\) is in the second quadrant, where the cosine value is negative. Therefore:

\[
\cos \theta = -\frac{\sqrt{35}}{6}
\]

Thus, the final answer is:

\[
\cos \theta = -\frac{\sqrt{35}}{6}
\]