If $$ \sec heta=-6 $$ and $$ \csc heta

Question

If $$ \sec 	heta=-6 $$ and $$ \csc 	heta<0 $$, find the exact value of $$ \sin 	heta $$

UpStudy AI · Accepted Answer

To find the exact value of $$ \sin \theta $$ given that $$ \sec \theta = -6 $$ and $$ \csc \theta < 0 $$, we start by recalling the definitions of the secant and cosecant functions:

$$
\sec \theta = \frac{1}{\cos \theta}
$$

From the given information, we have:

$$
\sec \theta = -6 \implies \cos \theta = -\frac{1}{6}
$$

Next, we need to determine the quadrant in which $$ \theta $$ lies. Since $$ \sec \theta $$ is negative, $$ \cos \theta $$ is also negative. The cosecant function $$ \csc \theta $$ is negative, which means that $$ \sin \theta $$ must also be negative. The only quadrant where both $$ \cos \theta < 0 $$ and $$ \sin \theta < 0 $$ is the third quadrant.

Now, we can use the Pythagorean identity to find $$ \sin \theta $$:

$$
\sin^2 \theta + \cos^2 \theta = 1
$$

Substituting $$ \cos \theta = -\frac{1}{6} $$:

$$
\sin^2 \theta + \left(-\frac{1}{6}\right)^2 = 1
$$

Calculating $$ \left(-\frac{1}{6}\right)^2 $$:

$$
\sin^2 \theta + \frac{1}{36} = 1
$$

Subtracting $$ \frac{1}{36} $$ from both sides:

$$
\sin^2 \theta = 1 - \frac{1}{36} = \frac{36}{36} - \frac{1}{36} = \frac{35}{36}
$$

Taking the square root of both sides, we find:

$$
\sin \theta = \pm \sqrt{\frac{35}{36}} = \pm \frac{\sqrt{35}}{6}
$$

Since we established that $$ \sin \theta < 0 $$ (as $$ \csc \theta < 0 $$), we conclude:

$$
\sin \theta = -\frac{\sqrt{35}}{6}
$$

Thus, the exact value of $$ \sin \theta $$ is:

$$
\boxed{-\frac{\sqrt{35}}{6}}
$$
The exact value of $$ \sin \theta $$ is $$ -\frac{\sqrt{35}}{6} $$.

If and ,
find the exact value of

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