1. If \(r \cos \theta = \frac{1}{2}\) and \(r \sin \theta = \frac{\sqrt{3}}{2}\), then find the value of \(r\), where \(0^\circ < \theta < 90^\circ\).

2. If \(2 \cos^2\theta + 3 \sin \theta = 3\) and \(0^\circ < \theta < 90^\circ\), find the value of \(\theta\).

3. Determine the values of \(\theta\) for which \(\sin^2\theta - 3\sin\theta + 2 = 0\) holds true, given that \(0^\circ < \theta < 90^\circ\).

4. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

5. If \( \csc \theta + \cot \theta = \sqrt{3} \), then find the value of \( \sin \theta \), where \( 0^\circ < \theta < 90^\circ \).

6. If \[ \tan \theta \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \] then find the value of \[ \sin(\theta - 15^\circ) \] given that \(0^\circ < \theta < 90^\circ\).

7. If \(5\sin^2 θ + 4\cos^2 θ = \frac{9}{2}\), then find the value of \(\tan θ\).

8. If \(\sin \theta + \sin^2 \theta = 1\), then find the value of \(\cos^2 \theta + \cos^4 \theta =\) _____

9. If \(\sec \theta = \csc \phi\), where \(0^\circ < \theta < 90^\circ\) and \(0^\circ < \phi < 90^\circ\), then the value of \(\sin(\theta + \phi)\) is 1.

10. If \(0^\circ < \theta \leq 90^\circ\), then what is the minimum value of \((4\csc^2\theta + 9\sin^2\theta)\)?

11. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

12. If \(\sin \theta = \cfrac{p^2 - q^2}{p^2 + q^2}\), then show that \(\cot \theta = \cfrac{2pq}{p^2 - q^2}\) where \(p > q\) and \(0^\circ < \theta < 90^\circ\).

13. Given: \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\) Prove that: \[ m^2 - n^2 = 4\sqrt{mn} \quad \text{where } 0^\circ < \theta < 90^\circ \]

14. If \(0^\circ < \theta < 90^\circ\), then determine the minimum value of \((9 \tan^2 \theta + 4 \cot^2 \theta)\).

15. If \(a \sin^2 \theta + b \cos^2 \theta = c\), then what is the value of \(\tan^2 \theta\)?

(a) \(\cfrac{b+c}{a+c}\) (b) \(\cfrac{b-c}{c-a}\) (c) \(\cfrac{a-b}{b-c}\) (d) \(\cfrac{c-a}{a-b}\)

16. If \( \csc A = \sqrt2 \), then find the value of
\(\cfrac{2\sin^2A + 3\cot^2A}{4\tan^2A - \cos^2A}\).

(a) \(\cfrac{8}{7}\) (b) \(\cfrac{7}{8}\) (c) \(\cfrac{1}{8}\) (d) \(\cfrac{1}{7}\)

17. Given \( r\cos\theta = 2\sqrt{3} \) and \( r\sin\theta = 2 \), where \( 0^\circ < \theta < 90^\circ \), find the values of \( r \) and \( \theta \).

18. Determine the value(s) of \(\theta\) within the range \(0^\circ \leq \theta \leq 90^\circ\) for which the equation \(2\cos^2\theta - 3\cos\theta + 1 = 0\) holds true.