Trigonometry: Mathematics 9709 (Cambridge International AS & A Level)

The graph of $y = 3 - 2\sin x$ is drawn for $0^\circ \le x \le 360^\circ$.

What is the minimum value of $y$?

For an angle $\theta$ where $\tan\theta$ is defined and non-zero, which expression is equivalent to $\dfrac{\sin\theta}{\tan\theta}$ for all such $\theta$?

(a) Show that the equation $2\sin^2 x = 1 + \cos x$ can be written in the form $2\cos^2 x + \cos x - 1 = 0$ [2]

(b) Hence solve $2\sin^2 x = 1 + \cos x$ for $0^\circ \le x \le 360^\circ$, giving all solutions. [4]

(a) Solve $2\cos 2x - 1 = 0$ for $0^\circ \le x \le 360^\circ$, giving all solutions. [5]

(b) Explain why part (a) has 4 solutions, rather than the 2 solutions that solving $\cos\theta = \tfrac12$ alone would suggest. [2]

(a) Show that the equation $2\sin x\tan x = 3$ can be written in the form $2\cos^2 x + 3\cos x - 2 = 0$ [3]

(b) Hence solve $2\sin x\tan x = 3$ for $0 \le x \le 2\pi$, giving your answers in terms of $\pi$. [3]

What is the exact value of $\sin 210°$?

The function $y = 5 - 4\cos(2x)$ is defined for $0° \le x \le 360°$.

(a) State the maximum value of $y$ and the minimum value of $y$. [2]

(b) Find all values of $x$ in the given interval at which the maximum value of $y$ occurs. [2]

(c) Find all values of $x$ in the given interval at which the minimum value of $y$ occurs. [2]

(a) Prove that $\sin\theta\tan\theta + \cos\theta \equiv \dfrac{1}{\cos\theta}$ for all $\theta$ where $\tan\theta$ is defined. [4]

(b) Hence, or otherwise, find the exact value of $\sin 60°\tan 60° + \cos 60°$. [2]

(a) Show that the equation $2\cos^2 x - 5\sin x = 4$ can be written in the form $2\sin^2 x + 5\sin x + 2 = 0$ [3]

(b) Hence solve $2\cos^2 x - 5\sin x = 4$ for $0 \le x \le 2\pi$, giving your answers in terms of $\pi$. [4]

The graph of $y = \sin(4x)$ is drawn for $x$ measured in degrees. What is the period of this graph?