Exercises: Double angle ratios

Knowing 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 $\alpha$ is in 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.

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

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

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 fully:

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

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

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

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

In the construction of an industrial building, two adjacent roofs have been designed. The first roof has an inclination $\alpha$ with respect 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 double 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 modeled by 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 located at a distance of $45 \text{ m}$, find the possible launch angles $\theta$ in degrees.

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

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

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

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

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

A ray of light is incident on a system of mirrors such that its final angle of deviation 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 $\cos(\theta)$.

Prove the following trigonometric identity, transforming the left-hand side until the right-hand side is obtained:

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

In the rhombus $ABCD$ in the figure, the diagonals measure $AC = 16 \text{ cm}$ and $BD = 12 \text{ cm}$. Let $\alpha$ be half 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 $\widehat{A}$.

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

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

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 $2\alpha$ with respect 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 located exactly on the Y-axis ($x_P = 0$).

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

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

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 $\left[0, \frac{\pi}{2}\right]$.

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 $[0^\circ, 360^\circ)$.

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 $\widehat{APB}$ which measures $\alpha$. Knowing that $\cos(\alpha) = \frac{4}{5}$, calculate the ratio (the quotient) between the area of the central triangle $\triangle OAB$ and the area of the inscribed triangle $\triangle PAB$.

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

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

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 time elapsed 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.