Exam-Style Questions.

1.	GCSE Higher

(a) Write down the exact value of \(\tan 60^o\).

(b) Find the exact area of this triangle.

Worked Solution

---

2.	IB Analysis and Approaches

Solve for x where \( -\pi \le x \le \pi \).

$$ \sin{2x} = \cos{x} $$

Worked Solution

---

3.	IB Analysis and Approaches

(a) Show that \(2x+15+\dfrac{40}{x-3}= \dfrac{2x^2+9x-5}{x-3}, \quad x \in \mathbb{R}, x \neq 3\)

(b) Hence or otherwise, solve the equation \( 2\cos{2\theta}+15+\dfrac{40}{\cos{2\theta}-3}=0, \quad \text{ for } 0 \le \theta \le \pi\)

Worked Solution

---

4.

A-Level

(a) Solve the following trigonometric equation for \(–360° \lt x \lt 360°\):

$$ 5 \sin^2 x + 2\sin x + 3 = 7 \cos^2 x $$

giving your answers to the nearest integer.

(b) Hence find the smallest positive solution of the equation

$$ 5 \sin^2(3\theta + 20°) + 2\sin (3\theta + 20°) + 3 = 7 \cos^2 (3\theta + 20°) $$

giving your answer to 2 decimal places.

Worked Solution

---

5.

A-Level

The cosine of acute angle \( \alpha \) is \( \frac{1}{ \sqrt 5} \)

The angle \( \beta \) is obtuse and \( \sin \beta = \sqrt \frac{2}{3} \).

(a) Find exact values of \( \tan \alpha \) and \( \tan \beta \).

(b) Hence show that \( \tan( \alpha - \beta ) \) can be written as \(a+b \sqrt 2 \) where \(a\) and \(b\) are rational numbers

Worked Solution

---