To solve the problem, we start by using the given information and trigonometric identities.

Part (a)

Given: $\sin(\theta) = \frac{\sqrt{14}}{19}$

Since $\theta$ is in the first quadrant, both sine and cosine are positive. The relationship between $\sin(\theta)$ and $\cos(\theta)$ is given by the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

Substituting the value of $\sin(\theta)$: $\left(\frac{\sqrt{14}}{19}\right)^2 + \cos^2(\theta) = 1$

Simplify the square of $\sin(\theta)$: $\frac{14}{361} + \cos^2(\theta) = 1$

Subtract $\frac{14}{361}$ from both sides: $\cos^2(\theta) = 1 - \frac{14}{361} = \frac{361}{361} - \frac{14}{361} = \frac{347}{361}$

Taking the square root of both sides: $\cos(\theta) = \frac{\sqrt{347}}{19}$

Since $\theta$ is in the first quadrant, $\cos(\theta)$ is positive, so: $\cos(\theta) = \frac{\sqrt{347}}{19}$

Part (b)

In the second quadrant, the sine of $\theta$ remains the same, but the cosine is negative. Thus, the exact value of $\cos(\theta)$ in the second quadrant is: $\cos(\theta) = -\frac{\sqrt{347}}{19}$

Final Answers:

• (a) $\cos(\theta) = \frac{\sqrt{347}}{19}$
• (b) $\cos(\theta) = -\frac{\sqrt{347}}{19}$

Would you like further explanation or details?

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Here are five related questions:

1. How would you find $\tan(\theta)$ in both the first and second quadrants?
2. What are the values of $\cos(\theta)$ and $\sin(\theta)$ if $\theta$ is in the third quadrant?
3. How does the Pythagorean identity change for angles in the third or fourth quadrants?
4. What is the significance of the sign of trigonometric functions in different quadrants?
5. How do you determine the exact value of $\sin(2\theta)$ given $\sin(\theta)$?

Tip: Remember, the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ is a powerful tool in trigonometry, especially for finding one trigonometric function given another.