Exercises: Trigonometric ratios of the difference of two angles

Calculate the exact value of $\cos(15^\circ)$ without using a calculator, applying the formula for the cosine of the difference of two angles and knowing that $15^\circ = 45^\circ - 30^\circ$.

Find the exact value of $\sin(15^\circ)$ by expressing the angle as the difference $60^\circ - 45^\circ$ and applying the corresponding identity.

Determine the exact value of $\tan(15^\circ)$ from the tangents of the special angles $45^\circ$ and $30^\circ$. Rationalise the result if necessary.

Simplify the expression $\sin(90^\circ - x)$ by applying the formula for the sine of the difference of two angles and verify that it is equivalent to one of the basic trigonometric ratios of the angle $x$.

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

Simplify the following trigonometric expression as much as possible: $\cos(x) \cos(y) + \sin(x) \sin(y)$. Then, use the simplified result to evaluate the expression exactly when $x = 75^\circ$ and $y = 15^\circ$.

Prove step by step that $\cos(180^\circ - \alpha) = -\cos \alpha$ using the formula for the cosine of the difference of two angles.

Calculate the exact value of $\tan(105^\circ)$ using the formula for the difference of two angles, considering that $105^\circ = 135^\circ - 30^\circ$.

Cian is solving an exercise where he has to calculate the exact value of $\cos(15^\circ)$ by applying the cosine difference formula with $15^\circ = 60^\circ - 45^\circ$. On his exam, he writes the following working:

When checking his result with a calculator, he is surprised to find that it gives a negative number (approximately $-0.26$), which he knows is impossible because $15^\circ$ lies in the first quadrant and its cosine must be positive.

1. a) Identify precisely the mistake Cian made and explain, based on the parity of trigonometric functions ($\cos(-x) = \cos x$ and $\sin(-x) = -\sin x$), why a central positive sign appears in the cosine difference formula.
2. b) Correct the calculation and find the exact, simplified value of $\cos(15^\circ)$.

You set out to calculate the exact value of $\sin(15^\circ)$ using the trigonometric ratios of special angles.

1. a) Before performing any analytical calculation, make a reasoned estimate of between which two simple decimal bounds $\sin(15^\circ)$ must lie: knowing that $\sin(0^\circ) = 0$ and $\sin(30^\circ) = 0.5$, and taking into account that the graph of the sine function on the interval $[0^\circ, 30^\circ]$ is concave down (it lies above the line segment joining the origin to the point $(30^\circ; 0.5)$), reason whether $\sin(15^\circ)$ must be strictly greater than, less than, or equal to $0.25$.
2. b) Decompose $15^\circ$ as $45^\circ - 30^\circ$ and apply the formula for $\sin(\alpha - \beta)$ to obtain the exact, simplified algebraic expression for $\sin(15^\circ)$.
3. c) Using the approximations $\sqrt{2} \approx 1.414$ and $\sqrt{6} \approx 2.449$, calculate the decimal value obtained to three decimal places and check whether your estimate from section a) was correct.

In a maritime navigation simulation, a ship must adjust its current heading. The necessary correction angle is given by the expression $\alpha - \beta$. Given that $\alpha$ and $\beta$ are acute angles with $\tan \alpha = 3$ and $\tan \beta = \frac{1}{2}$, calculate the exact value of $\tan(\alpha - \beta)$ and deduce the size of the correction angle.

Simplify the following algebraic trigonometric expression and express it purely in terms of the tangents of the angles $a$ and $b$:

$\frac{\sin(a - b)}{\cos a \cos b}$

A light ray passes through two different media with angles of refraction $\alpha$ and $\beta$. If, from the material properties, we know that $\cos \alpha = \frac{3}{5}$ (where $\alpha$ is an angle in the fourth quadrant) and $\sin \beta = \frac{5}{13}$ (where $\beta$ is an angle in the second quadrant), determine the exact value of $\cos(\alpha - \beta)$.

Evaluate the following trigonometric expression exactly, without using a calculator, by identifying which expansion it corresponds to:

$\sin(75^\circ) \cos(15^\circ) - \cos(75^\circ) \sin(15^\circ)$

Two coplanar forces are applied to an object forming angles $\alpha$ and $\beta$ with the horizontal, respectively. If $\sin \alpha = \frac{8}{17}$ and $\cos \beta = \frac{4}{5}$ (both angles being in the first quadrant), calculate the sine of the angle formed between both forces, that is, $\sin(\beta - \alpha)$.

Prove analytically that $\tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x}$. Use this result to find the exact value of $\tan(15^\circ)$ given that $\tan(30^\circ) = \frac{\sqrt{3}}{3}$.

Given that two angles $\alpha$ and $\beta$ are complementary ($\alpha + \beta = 90^\circ$), prove step by step that the identity $\sin(\alpha - \beta) = \cos(2\beta)$ holds.

In a market study, the seasonal sales cycles of two products are modelled using trigonometric functions. The phase difference that determines direct competition is given by the value of $\cos(x - y)$. If $\sin x = \frac{1}{3}$ and $\cos y = \frac{2}{5}$, with $x$ located in the first quadrant and $y$ in the fourth quadrant, find the exact value of $\cos(x - y)$.

The following data are known for two angles: $\sin \alpha = \frac{4}{5}$ with $\alpha \in (90^\circ, 180^\circ)$, and $\cos \beta = -\frac{5}{13}$ with $\beta \in (180^\circ, 270^\circ)$. You are asked to calculate the exact value of $\sin(\alpha - \beta)$. To solve it, two classmates propose different strategies:

• Method 1 (Direct): Calculate $\cos \alpha$ and $\sin \beta$ using the fundamental identity $\sin^2 x + \cos^2 x = 1$, choosing the appropriate sign according to the quadrant of each angle, and then apply the formula for $\sin(\alpha - \beta)$.

• Method 2 (Indirect): Calculate $\tan \alpha$ and $\tan \beta$, apply the tangent difference formula to find $\tan(\alpha - \beta)$, and from that value deduce $\sin(\alpha - \beta)$ using the relationship $1 + \tan^2(\alpha - \beta) = \frac{1}{\cos^2(\alpha - \beta)}$.

1. a) Fully carry out Method 1 and obtain the exact, simplified value of $\sin(\alpha - \beta)$.
2. b) Critically analyze Method 2: explain what fundamental ambiguity arises when attempting to determine the sign of $\sin(\alpha - \beta)$ solely from its tangent, and justify why Method 1 is more direct and reliable.

In Killarney National Park (Co. Kerry), a team from Mountain Rescue Ireland plans the evacuation of a hiker on a vertical cliff face. From the observation point in the valley, the line of sight to the top of the cliff forms an angle $\alpha$ with the horizontal such that $\tan \alpha = \frac{5}{2}$, while the line of sight to the base of the cliff forms an angle $\beta$ with the horizontal with $\tan \beta = \frac{3}{7}$ (both angles in the first quadrant). The total angle spanned by the cliff face from the observation post is $\theta = \alpha - \beta$.

1. a) Model the situation by applying the tangent difference formula to determine the exact, irreducible value of $\tan \theta = \tan(\alpha - \beta)$.
2. b) From the result obtained, deduce without a calculator the exact measure of the visual angle $\theta$ in sexagesimal degrees.
3. c) The rescue aeronautical protocol establishes that if the visual angle $\theta$ exceeds $40^\circ$, thermal turbulence from the cliff prevents a frontal approach and the helicopter must approach from the side flank. Reasonably justify which maneuver the pilot must order based on your calculation.

Solve the following trigonometric equation, finding all possible solutions in the interval $0^\circ \leq x < 360^\circ$:

$\sin(x - 30^\circ) = \frac{\sqrt{3}}{2}$

Prove the following trigonometric identity by appropriately combining the sum and difference angle formulas for sine and cosine:

$\frac{\cos(a - b) - \cos(a + b)}{\sin(a + b) + \sin(a - b)} = \tan b$

A surveyor observes a cliff of height $H$ and, located right on the edge above it, a lighthouse of height $h$. From their position at a horizontal distance $d$, the angle of elevation to the base of the lighthouse is $\beta$ and to the top of the lighthouse is $\alpha$. Express the tangent of the angle subtended by the lighthouse alone, that is $\tan(\alpha - \beta)$, purely in terms of the distances $H$, $h$ and $d$.

Solve the following system of trigonometric equations given that both $x$ and $y$ are angles belonging to the first quadrant:

$\sin x \cos y = \frac{3}{4}$

$\cos x \sin y = \frac{1}{4}$

Hint: First calculate the values of $\sin(x - y)$ and $\sin(x + y)$ by adding and subtracting both equations.

In the design of a robotic arm, the rotation angle of the end effector depends on the difference between two internal control angles $\alpha$ and $\beta$. If the tangent of the resulting angle is $\tan(\alpha - \beta) = \frac{1}{2}$ and due to mechanical constraints we know that $\tan \alpha = 3 \tan \beta$, determine the possible exact values of $\tan \beta$.

Prove analytically that $\sin(\alpha - \beta) \times \sin(\alpha + \beta) = \sin^2 \alpha - \sin^2 \beta$. Then, use this algebraic result to calculate mentally the exact value of the expression $\sin(75^\circ) \times \sin(15^\circ)$.

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

$\cos(x - 45^\circ) + \cos(x + 45^\circ) = 1$

In an alternating current circuit, the potential difference between two nodes is given by the superposition of two out-of-phase signals: $V(t) = 10 \sin(\omega t) - 10 \sin(\omega t - \frac{\pi}{3})$. Use the trigonometric difference angle formulas to express the potential difference $V(t)$ in the simplified form $A \sin(\omega t + \phi)$, determining the values of $A$ and $\phi$.

In the design of an optical sensor in a physics laboratory at the University College Cork, two light beams out of phase by acute angles $\alpha$ and $\beta$ (with $\alpha, \beta \in (0^\circ, 90^\circ)$) are required to satisfy the following tuning condition:

1. a) Expand both sides by applying, respectively, the formulas for the cosine of a difference and the sine of a sum of two angles.
2. b) Move all terms to the left-hand side and factor the resulting expression by grouping terms to prove that the above equality necessarily requires that $\tan \alpha = 1$ or $\tan \beta = 1$. What exact angular measures in degrees does this imply for $\alpha$ or $\beta$?
3. c) Justify why this result was foreseeable by relying exclusively on the property of trigonometric ratios of complementary angles ($\cos x = \sin(90^\circ - x)$) without needing to resort to the algebraic expansion of section a), and provide a specific pair of distinct acute angles $(\alpha, \beta)$ that satisfy the equality.

In an exhibition room of the National Gallery of Ireland (Dublin), a canvas measuring $3\text{ m}$ high hangs vertically on a wall. The bottom edge of the painting is located $2\text{ m}$ above the eye level of the visitors. An observer stands at a horizontal distance of $x$ meters from the wall ($x > 0$). Let $\alpha$ be the angle of elevation from the visitor's eyes to the top edge of the painting, and $\beta$ the angle of elevation to the bottom edge, so that the total visual angle under which the canvas is viewed is $\theta(x) = \alpha - \beta$.

1. a) Express $\tan \alpha$ and $\tan \beta$ in terms of the horizontal distance $x$.
2. b) Prove, by applying the formula for the tangent of the difference of two angles, that the tangent of the viewing angle is given by:

1. c) To maximize the view of the artwork's details, the distance $x$ that maximizes the angle $\theta(x)$ (and, consequently, $\tan \theta(x)$) is sought. Knowing that $\frac{3x}{x^2+10} = \frac{3}{x + \frac{10}{x}}$ and that the expression $x + \frac{10}{x}$ for $x > 0$ reaches its minimum when both terms are equal ($x = \frac{10}{x}$), calculate the exact distance in meters at which the visitor should stand to obtain the best perspective, and find the exact value of $\tan \theta$ at this optimal position.