Given that $\sin(\theta) = \frac{9}{\sqrt{430}}$, we need to find the exact value of $\cos(\theta)$ for both the first and second quadrants.

Step 1: Use the Pythagorean Identity

The Pythagorean identity states that: $\sin^2(\theta) + \cos^2(\theta) = 1$ Substitute $\sin(\theta)$ into the equation: $\left(\frac{9}{\sqrt{430}}\right)^2 + \cos^2(\theta) = 1$

Step 2: Calculate $\sin^2(\theta)$

$\left(\frac{9}{\sqrt{430}}\right)^2 = \frac{81}{430}$ So the equation becomes: $\frac{81}{430} + \cos^2(\theta) = 1$

Step 3: Solve for $\cos^2(\theta)$

$\cos^2(\theta) = 1 - \frac{81}{430}$ Simplify the right-hand side: $\cos^2(\theta) = \frac{430}{430} - \frac{81}{430} = \frac{349}{430}$

Step 4: Find $\cos(\theta)$

Taking the square root of both sides: $\cos(\theta) = \pm \sqrt{\frac{349}{430}}$

Now, we determine the sign of $\cos(\theta)$ based on the quadrant.

Case 1: Theta in the First Quadrant

In the first quadrant, $\cos(\theta)$ is positive. Therefore: $\cos(\theta) = \sqrt{\frac{349}{430}}$

Case 2: Theta in the Second Quadrant

In the second quadrant, $\cos(\theta)$ is negative. Therefore: $\cos(\theta) = -\sqrt{\frac{349}{430}}$

Final Answer:

• If $\theta$ is in the first quadrant, the exact value of $\cos(\theta)$ is $\sqrt{\frac{349}{430}}$.
• If $\theta$ is in the second quadrant, the exact value of $\cos(\theta)$ is $-\sqrt{\frac{349}{430}}$.

Do you want details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would you find $\tan(\theta)$ in both quadrants given $\sin(\theta)$?
2. What is the geometric interpretation of the Pythagorean identity on the unit circle?
3. What are the signs of all trigonometric functions in the second quadrant?
4. How does the value of $\theta$ affect the cosine function in different quadrants?
5. How can you derive the cosine function from a given sine function using trigonometric identities?

Tip: When solving trigonometric problems, always keep track of the quadrant to determine the correct sign of the function.