To find the decimal value of $\sin \theta$ given that $\cos \theta = \frac{7}{11}$, we can use the Pythagorean identity:

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

Substituting $\cos \theta = \frac{7}{11}$ into the equation:

$\sin^2 \theta + \left(\frac{7}{11}\right)^2 = 1$

First, calculate $\left(\frac{7}{11}\right)^2$:

$\left(\frac{7}{11}\right)^2 = \frac{49}{121}$

Now, substitute this back into the equation:

$\sin^2 \theta + \frac{49}{121} = 1$

Subtract $\frac{49}{121}$ from both sides:

$\sin^2 \theta = 1 - \frac{49}{121} = \frac{121}{121} - \frac{49}{121} = \frac{72}{121}$

Now, take the square root of both sides to find $\sin \theta$:

$\sin \theta = \sqrt{\frac{72}{121}} = \frac{\sqrt{72}}{11}$

Simplify $\sqrt{72}$:

$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

So:

$\sin \theta = \frac{6\sqrt{2}}{11}$

Now, find the decimal value:

$\sin \theta \approx \frac{6 \times 1.414}{11} \approx \frac{8.484}{11} \approx 0.771$

Thus, the decimal value of $\sin \theta$ is approximately 0.771.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the exact value of $\tan \theta$ given $\cos \theta = \frac{7}{11}$?
2. How does the Pythagorean identity help in solving trigonometric problems?
3. What are the possible signs of $\sin \theta$ based on the quadrant?
4. How can you find $\sec \theta$ given $\cos \theta$?
5. What are the other trigonometric identities that can be derived from $\sin^2 \theta + \cos^2 \theta = 1$?

Tip: Always consider the quadrant where the angle $\theta$ lies to determine the sign of trigonometric functions.