Double-Angle Ratios

Simplify the following trigonometric expression to a single exact numerical value:

$4\sin(15^\circ)\cos(15^\circ)$

Given that $\cos(x) = 0.6$, determine the value of $\cos(2x)$ using exclusively the identity relating the cosine of a double angle to the cosine of a single angle.

Given that $\sin(\alpha) = \frac{4}{5}$ and that $\alpha$ is an angle in the first quadrant, calculate the exact value of $\sin(2\alpha)$.

Given that $\cos(\alpha) = -\frac{1}{3}$ and that $\alpha$ belongs to the second quadrant, determine the exact value of $\cos(2\alpha)$.

If $\tan(\alpha) = \frac{3}{2}$, calculate the value of $\tan(2\alpha)$, expressing the result as an irreducible fraction.

A ray of light is incident on a mirror system such that its final deviation angle is $2\theta$. If laboratory measurements indicate that $\cos(2\theta) = 0.28$ and we know that the angle of incidence $\theta$ is acute, determine the exact value of $\sin(\theta)$ and $\cos(\theta)$.

In rhombus $ABCD$ shown in the figure, the diagonals measure $AC = 16 \text{ cm}$ and $BD = 12 \text{ cm}$. Let $\alpha$ be half of the interior angle at vertex $A$. Calculate the exact value of $\sin(2\alpha)$ and $\cos(2\alpha)$ to determine the sine and cosine of the full angle $\widehat{A}$.

Calculate the exact value of the following expression without using a calculator:

$\cos^2\left(\frac{\pi}{8}\right) - \sin^2\left(\frac{\pi}{8}\right)$

Write the following expression in terms of a single trigonometric ratio of the angle $x$, simplifying as much as possible:

$\frac{\sin(2x)}{2\sin(x)}$

Find the exact value of the following operation using double-angle identities:

$\frac{2\tan(22.5^\circ)}{1 - \tan^2(22.5^\circ)}$

Prove the following trigonometric identity, transforming the left-hand side to obtain the right-hand side:

$\frac{\sin(2x)}{1 + \cos(2x)} = \tan(x)$

A scissor mechanism for a hydraulic lift has a central pivot forming an opening angle $2\alpha$. The shear force on the central pin is given by an expression proportional to $F(\alpha) = \frac{1 - \cos(2\alpha)}{\sin(2\alpha)}$. Prove that this expression can be simplified to a single trigonometric ratio of the angle $\alpha$.

Prove the following trigonometric identity, valid for any angle $x$ where the expressions are defined:

$\frac{1 + \sin(2x) - \cos(2x)}{1 + \sin(2x) + \cos(2x)} = \tan(x)$

First prove the identity $\cos^4(x) - \sin^4(x) = \cos(2x)$. Then use this result to solve the equation $\cos^4(x) - \sin^4(x) = \frac{1}{2}$ in the interval $[0^\circ, 360^\circ)$.

Solve the following trigonometric equation, finding all solutions in the interval $[0, 2\pi)$:

$\sin(2x) + \sqrt{3}\cos(x) = 0$

Solve the following trigonometric equation and give the solutions in degrees for the interval $[0^\circ, 360^\circ)$:

$\cos(2x) - 3\sin(x) = 2$

Solve the following trigonometric equation, determining all solutions in the interval $[0, 2\pi)$:

$\tan(2x) \times \tan(x) = 1$

Solve the following system of trigonometric equations for $x, y \in [0, \pi/2]$:

$\begin{cases} \sin(x)\cos(y) + \cos(x)\sin(y) = 1 \\ \cos(2x) - \cos(2y) = 0 \end{cases}$

Find the values of the parameter $k \in \mathbb{R}$ for which the equation $\cos(2x) - 4\sin(x) + k = 0$ has exactly one solution in the interval $\left[0, \frac{\pi}{2}\right]$.

A company's revenue function from the sale of a seasonal product is given by the expression $I(x) = 5\sin(x)\cos(x) - 2\sin^2(x) + 1$, where $x$ is the elapsed time in months ($x \ge 0$). Express $I(x)$ in terms of $\sin(2x)$ and $\cos(2x)$, and determine the maximum revenue the company can achieve.

A planar robotic arm consists of two links, both of length $L$. The first link forms an angle $\alpha$ with the positive horizontal axis. The second link forms an angle $2\alpha$ to the horizontal. The $x$-coordinate of the end effector $P$ is given by the expression $x_P = L\cos(\alpha) + L\cos(2\alpha)$. Determine the exact value of the angle $\alpha \in [0, \pi/2]$ for which the end effector is situated exactly on the y-axis ($x_P = 0$).

In the construction of an industrial warehouse, two adjacent roofs have been designed. The first roof has an inclination $\alpha$ to the horizontal such that $\tan(\alpha) = 0.4$. For water drainage reasons, the second roof has been designed with an angle of inclination that is exactly twice that of the first, that is, $2\alpha$. Calculate the exact slope (tangent) of the second roof.

In physics, the maximum range $R$ of a projectile launched over horizontal ground is modelled using the formula $R = \frac{v_0^2 \sin(2\theta)}{g}$, where $v_0$ is the initial velocity, $\theta$ is the launch angle, and $g \approx 10 \text{ m/s}^2$ is the acceleration due to gravity. If a test cannon fires a projectile with an initial velocity of $30 \text{ m/s}$ and it is desired to hit a target situated $45 \text{ m}$ away, find the possible launch angles $\theta$ in degrees.

In a circle with center $O$ and radius $R$, the central angle $\widehat{AOB}$ measures $2\alpha$ and subtends the chord $AB$. A point $P$ located on the major arc subtends an inscribed angle $\widehat{APB}$ measuring $\alpha$. Given that $\cos(\alpha) = \frac{4}{5}$, calculate the ratio (quotient) between the area of the central triangle $\triangle OAB$ and the area of the inscribed triangle $\triangle PAB$.