1. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

2. If \((x + 1)\cot^2\frac{\pi}{2} = 2\cos^2\frac{\pi}{3} + \frac{3}{4}\sec^2\frac{\pi}{4} + 4\sin^2\frac{\pi}{6}\), then find the value of \(x\).

3. If \( α + β = \cfrac{\pi}{2} \), then find the value of \( \sqrt{\cfrac{\sinα}{\sinβ} - \sinα \cosβ} \).

(a) \(cosα\) (b) \(sinα\) (c) \(secα\) (d) \(cotα\)

4. If \(\tan 55^\circ \cdot \tan 35^\circ = \sin A\), then find the least positive value of \(A\).

(a) 0° (b) 35° (c) 55° (d) 90°

5. If \(\alpha + \beta = 90^\circ\) and \(\alpha : \beta = 2 : 1\), then find the value of \(\sin \alpha : \sin \beta\).

(a) \(3:1\) (b) \(1:3\) (c) \(\sqrt3:1\) (d) \(1:\sqrt 3\)

6. If \(\theta\) is a positive acute angle and \( \sin \theta - \cos \theta = 0 \), then the value of \(\cot 2\theta\) is –

(a) \(\cfrac{1}{√3}\) (b) 1 (c) √3 (d) 0

7. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

8. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

9. If \( \cot\alpha = \tan(\beta + \gamma) \), then what is the value of \( \sin(\alpha + \beta + \gamma) \)?

(a) 1 (b) 2 (c) 4 (d) \(\cfrac{3}{4}\)

10. If \(tan 35° \cdot tan 55° = \sin θ\), then the smallest positive value of \(θ\) will be ——.

11. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?

12. If \[ x \cot\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{3}\right) + \frac{3}{4} \sec^2\left(\frac{\pi}{4}\right) + 4 \sin\left(\frac{\pi}{6}\right) \] then find the value of \(x\).

13. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?

(a) 30° (b) 25° (c) 60° (d) 90°

14. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\). Let me know if you'd like a step-by-step solution next. I'm ready when you are.

15. 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.

16. If \(m = \sqrt{\frac{n}{n + b}}\), then express \(n\) in terms of \(m\) and \(b\).

17. If \( \cos 9\alpha = \sin \alpha \) and \( 9\alpha < 90^\circ \), then the value of \( \tan 5\alpha \) will be 1.

18. If \( \tan \theta \cdot \tan \phi = 1 \), then the value of \( \sin\left(\frac{\theta + \phi}{3}\right) \) will be –

(a) \(1\) (b) \(\cfrac{1}{\sqrt2}\) (c) \(\cfrac{\sqrt3}{2}\) (d) \(\cfrac{1}{2}\)

19. If \( \sin\alpha + \sin^2\alpha = 1 \), then what is the value of \( \cos^2\alpha + \cos^4\alpha \)?

20. If the base area of a right circular cone is \(A\), its height is \(H\), and its volume is \(V\), then express \(H\) in terms of \(V\) and \(A\).

(a) \(H=\cfrac{3V}{A}\) (b) \(H=\cfrac{V}{A}\) (c) \(H=\cfrac{V}{3A}\) (d) \(H=\cfrac{3V}{2A}\)

21. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

22. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

23. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

24. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).

25. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).

(a) 1 (b) 2 (c) 3 (d) 4

26. If \( \cos^2 θ - \sin^2 θ = \frac{1}{2} \), then find the value of \( \cos^4 θ - \sin^4 θ \).

27. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).

28. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \tan^2 θ \).

29. If \( \sin10θ = \cos8θ \) and \(10θ\) is a positive acute angle, find the value of \( \tan9θ \).

30. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).