Given $\cos(\theta) = -\frac{4}{5}$ and $\pi < \theta < \frac{3\pi}{2}$, we can determine $\sin(\theta)$ and $\tan(\theta)$ step-by-step.

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Step 1: Identify the quadrant

• Since $\pi < \theta < \frac{3\pi}{2}$, $\theta$ lies in the third quadrant.
• In the third quadrant:
  • $\sin(\theta)$ is negative.
  • $\cos(\theta)$ is negative.
  • $\tan(\theta)$ is positive.

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Step 2: Use the Pythagorean identity

The Pythagorean identity is:

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

Substitute $\cos(\theta) = -\frac{4}{5}$:

$\sin^2(\theta) + \left(-\frac{4}{5}\right)^2 = 1$

$\sin^2(\theta) + \frac{16}{25} = 1$

$\sin^2(\theta) = 1 - \frac{16}{25}$

$\sin^2(\theta) = \frac{25}{25} - \frac{16}{25}$

$\sin^2(\theta) = \frac{9}{25}$

$\sin(\theta) = \pm\frac{3}{5}$

Since $\theta$ is in the third quadrant, $\sin(\theta) < 0$, so:

$\sin(\theta) = -\frac{3}{5}$

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Step 3: Calculate $\tan(\theta)$

The formula for $\tan(\theta)$ is:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Substitute $\sin(\theta) = -\frac{3}{5}$ and $\cos(\theta) = -\frac{4}{5}$:

$\tan(\theta) = \frac{-\frac{3}{5}}{-\frac{4}{5}}$

$\tan(\theta) = \frac{3}{4}$

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Final Answers:

• $\sin(\theta) = -\frac{3}{5}$
• $\tan(\theta) = \frac{3}{4}$

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Would you like further clarification or a step-by-step visualization of the calculations? Here are five related questions to expand your understanding:

1. How can we determine the quadrant of an angle based on its cosine and sine values?
2. What happens to the trigonometric functions in different quadrants?
3. How is the Pythagorean identity used to relate sine and cosine?
4. What are some other ways to find the tangent of an angle using sine and cosine?
5. How would these calculations change if $\theta$ were in a different quadrant?

Tip: Always verify the sign of sine, cosine, and tangent using the quadrant rules!