Solving trig functions & quadratic equations with trig (GDC, analytically): notes and practice questions

Solve the following equations for $\theta$ in the interval $0 \le \theta \le 2\pi$. Give your answers as multiples of $\pi$.

(a) $\cos \theta = -\frac{\sqrt{3}}{2}$

(b) $\sin \theta = -\frac{1}{2}$

(c) $\tan \theta = -\sqrt{3}$

(a) Show that $2x+5 + \frac{3}{x-1} = \frac{2x^2 + 3x - 2}{x-1}$, for $x \in \mathbb{R}, x \neq 1$.

(b) Hence or otherwise, solve the equation $2\sin{\theta} + 5 + \frac{3}{\sin{\theta}-1} = 0$ for $0 \leq \theta \leq 2\pi$, $\theta \neq \frac{\pi}{2}$.

A ladder must be placed against a tall vertical building, clearing a monument that is 8 m high and stands on horizontal ground 1 m away from the building's base. The ladder touches the ground, the top corner of the monument, and the wall of the building.

Let $L$ be the length of the ladder in metres.

Let $\theta$ be the angle that the ladder makes with the ground, where $0 < \theta < \frac{\pi}{2}$.

(a) Show that $L = \sec \theta + 8\csc \theta$.

(b) (i) Find $\frac{dL}{d\theta}$.

(b) (ii) When $\frac{dL}{d\theta} = 0$, show that $\tan\theta = 2$.

(c) (i) Find $\frac{d^2L}{d\theta^2}$.

(c) (ii) When $\tan\theta = 2$, find the value of $\frac{d^2L}{d\theta^2}$.

(d) (i) Hence, justify that $L$ is a minimum when $\tan\theta = 2$.

(d) (ii) Determine this minimum value of $L$.

(e) A construction company only has ladders with a maximum length of 11 m. Determine whether it is possible to position a ladder against the building over the monument, giving a reason for your answer.

(a) Show that $\cos(2x) - \sin(x) = (1 - 2\sin(x) )(1 + \sin(x) )$.

(b) Hence, solve the equation $\cos(2x) - \sin(x) = \sin(x) + 1$ for $0 \le x < 2\pi$.

Consider the functions $f(x) = \sin x - \cos x$ and $h(x) = x + \frac{\pi}{4}$ for $x \in \mathbb{R}$.

(a) Find an expression for $(f \circ h)(x)$.

(b) Hence, solve the equation $(f \circ h)(x) = 1$ for $0 \le x \le 2\pi$.

(a) Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

(b) By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8 \cos^4\theta - 8 \cos^2\theta + 1$.

(c) Hence, find the four distinct roots of the equation $8x^4 - 8x^2 + 1 = 0$, expressing them in the form $\cos(\alpha)$ where $0 < \alpha < \pi$.

(d) By considering the roots of the equation in part (c), or otherwise, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$.

Hence, show that $\theta = \frac{\pi}{8}$ and $\theta = \frac{3\pi}{8}$ are solutions of the equation $8\cos^4\theta - 8\cos^2\theta + 1 = 0$.

Hence, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

Show that $1 - \cos(2x) - \sin(2x) = 2\sin x(\sin x - \cos x)$.

Hence, solve the equation $1 - \cos(2x) - \sin(2x) + \sin x - \cos x = 0$ for $0 < x < 2\pi$.

Show that the equation $2\sin^2x - \cos x - 1 = 0$ may be written in the form

$2\cos^2x + \cos x - 1 = 0$.

Hence, solve the equation $2\sin^2x - \cos x - 1 = 0$, for $0 \le x \le 2\pi$.

Consider the functions $f(x) = \sin x$ and $g(x) = \cos 2x$, where $0 \le x \le \pi$.

The graphs of $f$ and $g$ are shown in the following diagram.

The graphs intersect at points P and Q. The region enclosed by the two graphs is shaded and labelled R.

(a) Find the $x$-coordinates of P and Q.

(b) Find the area of R.

Show that $3x + 2 - \frac{2}{x+1} = \frac{3x^2 + 5x}{x+1}$, for $x \in \mathbb{R}, x \neq -1$.

Hence or otherwise, solve the equation $3\cos(\theta) + 2 - \frac{2}{\cos(\theta)+1} = 0$ for $0 \le \theta \le 2\pi, \theta \neq \pi$.

Expand and simplify $(1+a)^3$ in ascending powers of $a$.

By using a suitable substitution for $a$, show that $1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) = 8\cos^6(x)$.

Consider the function $g(x) = 4\sin(x)(1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) )$.

Show that $\int_0^p g(x) \, dx = \frac{32}{7}(1-\cos^7 p)$, where $p$ is a positive real constant.

It is given that $\int_p^{\frac{\pi}{2}} g(x) \, dx = \frac{1}{28}$, where $0 \le p \le \frac{\pi}{2}$. Find the value of $p$.

Prove that $\cot\left(\theta + \frac{\pi}{4}\right) = \frac{1 - \sin 2\theta}{\cos 2\theta}$, where $\theta \ne \frac{\pi}{4} + \frac{n\pi}{2}, n \in \mathbb{Z}$.

Hence, solve the equation $\frac{1 - \sin x}{\cos x} = 1$ for $0 \le x \le 2\pi$.

Consider the functions $f(x) = \sin x + \sqrt{3}\cos x$ and $g(x) = 2x$.

(a) Find $(f \circ g)(x)$.

(b) Solve the equation $(f \circ g)(x) = 2\sin(2x)$ for $0 \leq x \leq \pi$.

A team of engineers is testing two autonomous robots, Alpha and Beta, on a straight track. Their positions are measured as the distance from a fixed starting point. The experiment runs for 10 minutes.

The position of Robot Alpha, $P_A$ metres, at time $t$ minutes can be modelled by the function $P_A(t) = 3\sin(2t + 5) + 14t + 20$, where $0 \le t \le 10$.

The position of Robot Beta, $P_B$ metres, at time $t$ minutes can be modelled by the function $P_B(t) = 12t + 25$, where $0 \le t \le 10$.

Use the engineers' models to find the initial position of

(i) Robot Beta;

(ii) Robot Alpha correct to three significant figures.

Find the values of $t$ when Robot Alpha and Robot Beta are at the same position. Give your answers correct to three significant figures.

For $t > 5$, prove that Robot Alpha was always ahead of Robot Beta.

For $0 \le t \le 10$, find the total amount of time when the speed of Robot Beta was greater than the speed of Robot Alpha. Give your answer correct to three significant figures.