Exercises: Trigonometric ratios of the sum of two angles

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:

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

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

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.

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 $\cos(105^\circ)$.

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 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 $\cos(\alpha + \beta)$.

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

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

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 $\tan(15^\circ) = 2 - \sqrt{3}$.

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.

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.

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, $\sin(\alpha + \beta)$.

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 $30^\circ$.

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 $45^\circ$.

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 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 $[0, 2\pi)$.

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.

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 $\tan \beta$.

In any non-right triangle $ABC$, given that the sum of its interior angles is $A+B+C = 180^\circ$, mathematically prove the following fundamental relation:

$\tan A + \tan B + \tan C = \tan A \tan B \tan C$

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.

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:

$\sin(x+y) = 1$

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

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.

To solve in the interval $[0^\circ, 360^\circ)$ the trigonometric equation:

two students propose different solving methods:

• Aoife's method: Directly applies the addition formulas for the sine and cosine of the sum to the left-hand side, combines like terms, and solves the resulting simple equation.

• Cian's method: Uses the substitution $u = x + 45^\circ$, squares both sides of the equation $\sin u + \cos u = 1$ to get $(\sin u + \cos u)^2 = 1$, and uses the fundamental identity to solve for $u$ and then for $x$.

1. a) Carry out Aoife's method completely and find all exact solutions of the equation in the interval $[0^\circ, 360^\circ)$.
2. b) Carry out Cian's method until finding the apparent solutions for $x$. Explain in detail why squaring introduces extraneous solutions in this context and identify which ones they are by checking them in the original equation.
3. c) Justify why Aoife's method based on the addition formulas is conceptually superior and safer than resorting to squaring.

In any non-right triangle $ABC$, the measures of its interior angles satisfy $A + B + C = 180^\circ$. In a specific case, the tangents of two of its angles are known: $\tan A = 2$ and $\tan B = 3$.

1. a) Knowing that $C = 180^\circ - (A + B)$, apply the properties of supplementary angles to prove the relationship between $\tan C$ and $\tan(A + B)$.
2. b) Apply the angle addition formula for tangent to calculate the exact value of $\tan(A+B)$ and deduce the numerical value of $\tan C$. From this, find the exact measure in degrees of angle $C$ and reason whether the triangle is acute or obtuse.
3. c) Check numerically whether for this triangle the equality $\tan A + \tan B + \tan C = \tan A \times \tan B \times \tan C$ holds. Next, prove algebraically that this relation is an identity that holds for any non-right triangle starting from the equality $\tan(A+B) = -\tan C$.