Trig identities (pythagorean identity, double angle) & relationships: notes and practice questions

$a$ Find $\int_{}^{}{\cos^{2}(x)}dx$.

$b$ Hence, evaluate $\int_{0}^{\frac{\pi}{2}}{\cos^{2}(x)}dx$.

$a$ Find $\int_{}^{}{\cos^{2}(x)}dx$.

$b$ Hence, evaluate $\int_{0}^{\frac{\pi}{2}}{\cos^{2}(x)}dx$.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

(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 $0 \le x < 2\pi$.

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) 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 $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

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 $0 \le x \le 2\pi$.

Consider the function $g(x)=\frac{1-\cos(ax)}{x^2}$, where $x \neq 0$ and $a \in \mathbb{R}^+$.

(a) Show that $g$ is an even function.

(b) Given that $\lim_{x\to0} g(x) = 8$, find the value of $a$.

Consider the functions $f(x) = -3\cos x + 5$ and $g(x) = -3\cos\left(x+\frac{\pi}{2}\right) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, p)$.

Given that the maximum value of $g(x)$ is less than or equal to 1, find the largest possible value of $p$.

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 $0 < x < 2\pi$.

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.

A curve is given by the equation $e^y = \cos(x)$ for $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$.

(a) Use implicit differentiation to show that $\frac{dy}{dx} = -\tan(x)$.

(b) Show that $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 + 1 = 0$.

(c) Find an expression for $\frac{d^3y}{dx^3}$ in terms of $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.

(d) Hence, find the Maclaurin series for $y = \ln(\cos(x) )$ up to and including the term in $x^4$.