If cosθ = 7/11 If find the decimal value of sinθ

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.

Learn how to find the decimal value of sin θ when cos θ = 7/11 using trigonometric identities and calculations. Understand the Pythagorean identity and its application to solve trigonometric problems. Suitable for high school students and those preparing for trigonometry exams.

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