Trigonometric ratios of the sum of two angles

Express $\cos(x + \frac{\pi}{2})$ in terms of the trigonometric ratios of the angle $x$ by expanding the corresponding sum formula and simplifying the result.

Simplify the trigonometric expression $\cos(x + \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$ so that it is in terms of $\cos x$ and/or $\sin x$ only.

Expand the expression $\cos(\alpha+\beta+\gamma)$ purely in terms of the sines and cosines of the single angles $\alpha$, $\beta$ and $\gamma$. Group the terms appropriately.

Aoife needs to calculate the exact value of $\cos(105^\circ)$ without a calculator and decides to decompose the angle as the sum of two special angles: $105^\circ = 60^\circ + 45^\circ$. In her exam, she writes the following working:

When checking her graded test, she finds that the teacher gave the exercise zero marks.

1. a) Without performing analytical calculations, reason geometrically using the position on the unit circle why Aoife's result ($\approx 0.97$) is impossible for an angle of $105^\circ$.
2. b) Locate and explain precisely the conceptual or formula error made in the second equality of her working.
3. c) Correct the procedure and obtain the simplified exact value of $\cos(105^\circ)$.

Given two angles $\alpha$ and $\beta$ such that $\tan \alpha = 3$ and $\tan \beta = 2$, determine the exact value of $\tan(\alpha + \beta)$.

Calculate the exact value of $\sin(75^\circ)$ by decomposing the angle as the sum of two special angles ($45^\circ$ and $30^\circ$).

Determine the exact value of $\cos(105^\circ)$ using the formula for the cosine of a sum, given that $105^\circ = 60^\circ + 45^\circ$.

Find the exact value of $\tan(75^\circ)$ by applying the formula for the tangent of a sum using the angles of $45^\circ$ and $30^\circ$. Rationalise the result if necessary.

Simplify the following trigonometric expression to a single exact numerical term by applying the addition formulas in reverse:

$\sin(22^\circ)\cos(8^\circ) + \cos(22^\circ)\sin(8^\circ)$

Reduce the following expression to an exact numerical value:

$\cos(40^\circ)\cos(20^\circ) - \sin(40^\circ)\sin(20^\circ)$

Calculate the exact value of the expression $\sin(15^\circ)\cos(30^\circ) + \cos(15^\circ)\sin(30^\circ)$. Use this geometric result to find the exact area of a parallelogram whose adjacent sides measure $8$ cm and $10$ cm, given that the angle they form is precisely the sum of $15^\circ$ and $30^\circ$.

Prove the following trigonometric identity by expanding the numerator using the corresponding formula and simplifying the fraction:

$\frac{\sin(\alpha+\beta)}{\cos\alpha \cos\beta} = \tan\alpha + \tan\beta$

The aim is to construct two acute angles $\alpha$ and $\beta$ belonging to the first quadrant, $\alpha, \beta \in (0^\circ, 90^\circ)$, that simultaneously satisfy these two conditions:

1) $\cos\alpha\times\cos\beta - \sin\alpha\times\sin\beta = 0$

2) $\tan\alpha = 3\times\tan\beta$

1. a) Recognize the trigonometric formula corresponding to the left-hand side of condition (1) and deduce with reasoning what complementary relationship $\alpha$ and $\beta$ hold with each other.
2. b) Using this relationship, express $\tan\beta$ exclusively in terms of $\tan\alpha$ using the trigonometric ratios of complementary angles.
3. c) Substitute the previous result into condition (2), solve the resulting equation to find the exact value of $\tan\alpha$, and determine the exact measures in degrees of both angles $\alpha$ and $\beta$.

Solve the following system of trigonometric equations for $x, y \in [0, \frac{\pi}{2}]$ and give the exact values of $x$ and $y$ in radians:

$\sin(x+y) = 1$

$\cos x \sin y = \frac{1}{2}$

Given that $\sin \alpha = \frac{4}{5}$ and $\cos \beta = \frac{12}{13}$, and that both $\alpha$ and $\beta$ are acute angles (belonging to the first quadrant), calculate the exact value of $\sin(\alpha + \beta)$.

Given the angles $\alpha$ and $\beta$ such that $\sin \alpha = \frac{3}{5}$ with $\alpha \in [\frac{\pi}{2}, \pi]$ and $\cos \beta = \frac{5}{13}$ with $\beta \in [\frac{3\pi}{2}, 2\pi]$, calculate the exact value of $\cos(\alpha + \beta)$.

A cannon fires a projectile at an angle of elevation $\alpha$ relative to the slope of a mountain, satisfying $\sin \alpha = 0.6$ (where $\alpha$ is an acute angle). The slope itself has an inclination $\beta$ relative to the horizontal with $\cos \beta = 0.96$ (where $\beta$ is acute). Find the exact value of the sine of the total launch angle relative to the horizontal, that is, $\sin(\alpha + \beta)$.

A surveyor observes the roof of a building at an angle of elevation $\alpha$, satisfying $\cos \alpha = \frac{4}{5}$. Behind this building, in the same line of sight, there is a telecommunications tower. The additional angle of elevation from the roof of the building to the top of the tower, measured from the previous line of sight, is $\beta$, with $\sin \beta = \frac{7}{25}$. Given that both angles are acute, calculate the exact value of the cosine of the total angle of elevation $\alpha + \beta$ towards the top of the tower.

Consider two unknown acute angles, $\alpha$ and $\beta$, for which it is only known that $\tan\alpha = 2$ and $\tan\beta = 3$.

1. a) Estimate: Taking into account that in the first quadrant the tangent is a strictly increasing function and that $\tan(45^\circ) = 1$, reason which quadrant the sum angle $\alpha + \beta$ must lie in. From this qualitative deduction, predict what sign (+ or -) $\tan(\alpha+\beta)$ will necessarily have before doing any calculations.
2. b) Calculate: Apply the angle addition formula for tangent to find the exact value of $\tan(\alpha+\beta)$.
3. c) Check: Assess whether the obtained numerical value confirms your sign prediction and determine the exact measure in degrees of the angle $\alpha + \beta$.

Given that $\tan \alpha = \frac{1}{2}$ and that the tangent of the sum angle is $\tan(\alpha + \beta) = 3$, determine the exact value of $\tan \beta$ by setting up the corresponding equation from the addition formula.

In an optics experiment, a light beam passes through two consecutive prisms that deviate its path in the same plane. The first prism produces a deviation $\alpha$ such that $\tan \alpha = \frac{1}{3}$, and the second an additional deviation $\beta$ such that $\tan \beta = \frac{1}{2}$. Prove that the total deviation $\alpha + \beta$ is exactly $45^\circ$ by calculating the tangent of the sum of the angles.

In a photovoltaic installation located in Albacete, the modules are placed on a natural slope that has an upward inclination of acute angle $\beta$ with respect to the horizontal, satisfying $\tan\beta = \frac{7}{24}$. The mounting brackets allow the panels to be tilted by an additional acute angle $\alpha$ with respect to the ground of the slope, such that the total inclination above the horizontal plane is $\alpha + \beta$. To optimize solar capture during the winter months, the technical specifications require that the total angle satisfies $\sin(\alpha + \beta) = \frac{4}{5}$, with $\alpha + \beta < 90^\circ$.

1. a) From $\tan\beta = \frac{7}{24}$, obtain the exact values of $\sin\beta$ and $\cos\beta$, justifying the sign according to the quadrant.
2. b) Determine the exact value of $\cos(\alpha + \beta)$ and calculate from it the exact fraction representing $\tan(\alpha + \beta)$.
3. c) Use the angle addition formula for tangent to set up a linear equation and calculate the exact value of $\tan\alpha$. From this result, reason whether the mechanical adjustment $\alpha$ on the terrain must be greater or less than $45^\circ$.

In a solar engineering project, a photovoltaic panel has an initial inclination $\alpha$ such that $\tan \alpha = \frac{3}{4}$. To optimise collection in winter, a mechanism increases the inclination by an additional angle of $15^\circ$. Calculate the exact value of $\tan(\alpha + 15^\circ)$ without using decimal approximations, given that $\tan(15^\circ) = 2 - \sqrt{3}$.

Solve the trigonometric equation $\sin(x + 30^\circ) = 2\cos x$ for the interval $x \in [0^\circ, 360^\circ)$. Express the solutions exactly in degrees.

Find all exact solutions to the equation $\sin(x + \frac{\pi}{6}) - \cos x = 0$ in the interval $[0, 2\pi)$.

A planar robotic arm consists of two links. The first link forms an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac{\sqrt{2}}{2}$ (lying in the first quadrant). The second link forms an angle $\beta$ of relative rotation with respect to the extension of the first link. If reaching a critical part requires the total inclination of the second link with respect to the horizontal to be such that $\tan(\alpha + \beta) = 2$, calculate the exact value of $\tan \beta$.

A reconnaissance aircraft initially flies on a heading forming an angle $\alpha$ north of east, satisfying $\tan \alpha = \frac{3}{4}$. A strong crosswind deflects its trajectory by an additional angle $\beta$ towards the north, so that the effective ground track forms an angle $\alpha + \beta$ with the east. If the control tower measures that the tangent of the effective heading is $\frac{12}{5}$, find the exact values of $\sin \beta$ and $\cos \beta$ without calculating the angles using a calculator.