Problems adapted from questions set for previous Mathematics exams.
(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.
The function \(f\) is defined as \(f(x) = 12x^3 - 5x^2 -11x + 6 \).
(a) Use the Factor Theorem to show that \( (4x-3) \) is a factor of \(f(x)\)
(b) Express \(f(x)\) as a product of linear factors.
(d) Hence solve the equation \( g(\theta ) = 0 \), giving your answers, in radians, in the interval \(0 \le \theta \le 2 \pi \).