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Let's go through each part of the problem and solve it:

### (i) Given \( \sin \theta = -\frac{3}{5} \), with \( 180^\circ \leq \theta \leq 270^\circ \), find \( \cos \theta \).

- First, recognize that \( \theta \) is in the **third quadrant** (since \( 180^\circ \leq \theta \leq 270^\circ \)).
- In the third quadrant, \( \sin \theta \) is negative, and \( \cos \theta \) is also negative.

- Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we use the identity:
  \[
  \sin^2 \theta + \cos^2 \theta = 1
  \]
  Substituting \( \sin \theta = -\frac{3}{5} \):
  \[
  \left( -\frac{3}{5} \right)^2 + \cos^2 \theta = 1
  \]
  \[
  \frac{9}{25} + \cos^2 \theta = 1
  \]
  \[
  \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}
  \]
  \[
  \cos \theta = \pm \frac{4}{5}
  \]
  Since \( \cos \theta \) is negative in the third quadrant, we have:
  \[
  \cos \theta = -\frac{4}{5}
  \]

### (ii) Determine the quadrant where \( \sin \theta < 0 \) and \( \tan \theta > 0 \).

- If \( \sin \theta < 0 \), then \( \theta \) is either in the **third** or **fourth quadrant**.
- If \( \tan \theta > 0 \), then \( \theta \) is in a quadrant where both sine and cosine have the same sign. This happens in the **third quadrant**, where both sine and cosine are negative, and their ratio (tangent) is positive.

Thus, \( \theta \) lies in the **third quadrant**.

### (iii) If \( f(x) = x^2 + 2x - 1 \), find \( f(0) \).

- To find \( f(0) \), substitute \( x = 0 \) into the function:
  \[
  f(0) = (0)^2 + 2(0) - 1 = -1
  \]
Thus, \( f(0) = -1 \).

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Would you like further clarification or details on any of these parts?

Here are 5 related questions for practice:
1. Find \( \cos \theta \) if \( \sin \theta = \frac{5}{13} \) and \( \theta \) is in the second quadrant.
2. In which quadrant is \( \theta \) if \( \sin \theta > 0 \) and \( \cos \theta < 0 \)?
3. If \( \tan \theta = -1 \), in which quadrant is \( \theta \)?
4. For \( f(x) = 3x^2 - 5x + 4 \), find \( f(2) \).
5. Solve for \( x \) if \( f(x) = 0 \) for the function \( f(x) = x^2 - 6x + 9 \).

**Tip:** When using trigonometric identities like \( \sin^2 \theta + \cos^2 \theta = 1 \), always consider the quadrant to determine the correct sign of the values.

If sin(θ) = -3/5, 180° ≤ θ < 270°, find cos(θ). In which quadrant does θ lie if sin(θ) < 0 and tan(θ) > 0? If f(x) = x^2 + 2x - 1, find f(0).

This problem involves finding cos(θ) given sin(θ) and determining the quadrant, as well as solving a quadratic function. Explore solutions using key trigonometric identities and function evaluation. Suitable for Grades 10-12.

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