Exercises: Trigonometric ratios of an angle

Convert the angles of $135^\circ$ and $330^\circ$ to radians, and express the angles of $\frac{5\pi}{6}$ rad and $\frac{7\pi}{4}$ rad in sexagesimal degrees.

Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right-angled triangle, knowing that its shorter sides measure $5$ cm and $12$ cm.

Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using fundamental trigonometric identities.

Evaluate the following expression by replacing the trigonometric ratios with their exact values, without using a calculator:

$\sin^2(30^\circ) + \cos^2(45^\circ) - \tan(60^\circ) \cdot \sin(60^\circ)$

Determine which quadrant the angle $\beta$ lies in if both $\sin \beta < 0$ and $\tan \beta > 0$ hold simultaneously.

Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and cotangent.

In a right-angled triangle, the tangent of one of its acute angles is $\frac{3}{4}$. Determine the exact value of the secant and cosecant of this angle.

Calculate the numerical value of the following trigonometric expression corresponding to angles located on the semi-axes:

$3\sin(90^\circ) - 2\cos(180^\circ) + 4\sin(270^\circ) - \cos(360^\circ)$

Lucía has to calculate the trigonometric ratios of the angle $\alpha$, knowing that it belongs to the second quadrant and that $\sin\alpha = \frac{3}{5}$. On her exam, she writes:

and then concludes that $\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{3/5}{4/5} = \frac{3}{4}$.

1. a) Point out precisely the conceptual error made by Lucía and explain, relying on the unit circle, why her result for the sign of $\cos\alpha$ and $\tan\alpha$ is inconsistent with the indicated quadrant.
2. b) Correct the solution and calculate the exact numerical value of $\cos\alpha$ and $\tan\alpha$.

Consider the angle $\alpha = 210^\circ$.

1. a) Without using a calculator or performing prior analytical calculations, make a reasoned estimate by placing the angle on the unit circle: indicate which quadrant it belongs to, deduce the sign of $\sin 210^\circ$, $\cos 210^\circ$, and $\tan 210^\circ$, and justify which of the two quantities in absolute value must be greater, $|\sin 210^\circ|$ or $|\cos 210^\circ|$, based on whether the radius points closer to the horizontal axis or the vertical axis.
2. b) Now deduce the exact numerical values of $\sin 210^\circ$, $\cos 210^\circ$, and $\tan 210^\circ$ by relating this angle to its corresponding special angle in the first quadrant ($30^\circ$).
3. c) Check whether the exact values you obtained confirm the estimate of signs and relative magnitudes formulated in part a).

Given that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.

A building casts a shadow of $24$ metres when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height of the building and determine its value. If the angle of elevation is halved, does the length of the shadow double? Justify your answer based on the non-linearity of the tangent ratio.

Simplify the following trigonometric expression as much as possible using fundamental identities, expressing the result in terms of a single trigonometric ratio:

$\frac{\sin^3 x + \sin x \cdot \cos^2 x}{\cos x}$

A grinder disc rotates at $7200$ revolutions per minute. Express this angular velocity in radians per second and calculate the exact angle (in radians) swept out by a particle on its outer edge in $0,15$ seconds.

Find the exact value of the following expression by first reducing all the angles to the first quadrant:

$\cos(120^\circ) \cdot \sin(225^\circ) - \tan(330^\circ) \cdot \cos(210^\circ)$

A pendulum of length $80$ cm moves from its vertical equilibrium position to form an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance that the pendulum bob has moved and the height it has risen with respect to its lowest point. Express the results in terms of the trigonometric ratios of $18^\circ$.

Calculate the exact values of $\cot \alpha$ and $\csc \alpha$ given that $\sec \alpha = 3$ and that the sine of the angle is negative.

To optimise solar energy collection in winter, a rectangular panel $2$ metres long is raised by resting one of its shorter sides on flat ground, so that the opposite side is at a height of exactly $1,2$ metres. Determine the exact value of the cosine and tangent of the angle of inclination of the panel.

It is known that an angle $\alpha$ belongs to the second quadrant and that $\tan\alpha = -\frac{4}{3}$. To calculate the exact values of $\cos\alpha$ and $\sin\alpha$, two classmates propose two different strategies:

• Method A (Raquel): Applies the identity $1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}$, solves for $\cos\alpha$ choosing the appropriate sign for the quadrant, and then obtains $\sin\alpha = \tan\alpha \cdot \cos\alpha$.

• Method B (Sergio): Sets up a system of equations with the definition $\frac{\sin\alpha}{\cos\alpha} = -\frac{4}{3}$ and the fundamental Pythagorean identity $\sin^2\alpha + \cos^2\alpha = 1$, expressing one unknown in terms of the other to substitute.

1. a) Compare both methods, analyzing which one is more direct, which involves less algebraic computation, and in which one it is easier to make a mistake with the choice of signs.
2. b) Apply the method you consider most efficient to find the exact values of $\cos\alpha$ and $\sin\alpha$, justifying the choice of sign at each step based on the quadrant of $\alpha$.

Write a quadratic polynomial equation in the unknown $\cos x$, of the form $a\cos^2 x + b\cos x + c = 0$ with integer coefficients ($a, b, c \in \mathbb{Z}$ and $a > 0$), whose solutions in the interval $[0^\circ, 360^\circ)$ are solely and exclusively the four angles:

To construct and justify it:

1. a) Determine the exact numerical value that $\cos x$ takes at each of these four angles using the symmetry of the unit circle.
2. b) Construct the equation in terms of $\cos x$ from these values and simplify its coefficients so that they are integers.
3. c) Justify why your equation guarantees that there is no other angle in the first full rotation $[0^\circ, 360^\circ)$ that satisfies the equality.

Prove the following trigonometric identity, showing the algebraic steps and fundamental relations used. Assume that $\alpha$ is an angle for which all ratios are defined:

$\frac{1 - \sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{1 - \sin \alpha} = 2\sec \alpha$

Prove that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression $A = \frac{n \cdot R^2}{2} \sin\left(\frac{360^\circ}{n}\right)$. Then, apply the formula to calculate the exact area of a regular octagon inscribed in a circle of radius $10$ cm.

Given the relation $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasons which quadrants the angle $\theta$ can lie in.

Simplify the following trigonometric expression as much as possible, assuming that $x$ is an angle in the first quadrant. Pay special attention to the signs when reducing to that quadrant:

$\frac{\sin(\pi - x) \cdot \cos\left(\frac{\pi}{2} + x\right) \cdot \tan(2\pi - x)}{\cos(\pi + x) \cdot \sin\left(\frac{3\pi}{2} - x\right)}$

Find all angles $\alpha \in [0^\circ, 360^\circ)$ satisfying the following trigonometric equation:

$2\cos^2 \alpha - 3\sin \alpha = 0$

(Hint: express the entire equation in terms of a single trigonometric ratio using the fundamental identity).

A light ray passes from a liquid medium with refractive index $n_1 = 1,33$ to a crystal with index $n_2 = 1,52$. Snell's law states that $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction measured relative to the normal to the surface. If the ray is incident at an angle of $\theta_1 = \frac{\pi}{4}$ rad, determine the exact value of the cosine of the angle of refraction $\theta_2$.

Two circular pulleys of radii $R = 15$ cm and $r = 5$ cm are connected by a taut belt crossed between them. The distance between the centres of the pulleys is $40$ cm. If $\alpha$ is the angle formed by the straight section of the belt with the imaginary line joining the centres, calculate the exact value of $\sin \alpha$, $\cos \alpha$ and the total length of the straight section of the belt between the two points of tangency.

Prove that if $\alpha$ and $\beta$ are two acute complementary angles (that is, $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relation always holds:

$\frac{\tan \alpha + \tan \beta}{\sec \alpha \cdot \sec \beta} = 1$

In a wind farm in Castilla y León, a wind turbine has its hub (center of rotation of the blades) located at $90\text{ m}$ above the ground and its blades are $50\text{ m}$ long. The position of the tip of a blade is modeled by a circle of radius $50\text{ m}$ centered at $(0, 90)$, where the angle $\alpha$ is measured counterclockwise from the right horizontal position ($3$ o'clock on a watch).

Due to a wind gust, the control system locks the blade when the angle $\alpha$ is in the second quadrant and the anemometer records that $\tan\alpha = -\frac{7}{24}$.

1. a) Deduce, using fundamental trigonometric identities and justifying the sign according to the quadrant, the exact values of $\cos\alpha$ and $\sin\alpha$.
2. b) Calculate the exact coordinates $(x, y)$ of the blade tip relative to the ground (taking the origin at the base of the vertical mast), indicating its height above the ground and its horizontal displacement.
3. c) Relying on the symmetry properties of the unit circle, determine what other angle $\beta \in [0^\circ, 360^\circ)$ would place the blade tip at exactly the same height above the ground, and calculate the exact value of $\sin\beta$, $\cos\beta$, and $\tan\beta$.

On a written test, students are asked to simplify the following expression for all angles $\alpha \in [0^\circ, 360^\circ)$ for which it is defined:

Carlos turns in the following answer:

"Since $\sin^2\alpha + \cos^2\alpha = 1$, it holds that $\sqrt{1 - \sin^2\alpha} = \cos\alpha$ and that $\sqrt{1 - \cos^2\alpha} = \sin\alpha$. Substituting into the expression:

Therefore, the result is $0$ for any angle where it exists."

1. a) Explain with algebraic rigor the error made by Carlos when taking the square roots and state the general property $(\sqrt{u^2})$ that he omitted.
2. b) State, with reasoning, for which angles $\alpha \in [0^\circ, 360^\circ)$ the expression $E(\alpha)$ is not defined.
3. c) Perform a quadrant-by-quadrant study of the constant value that $E(\alpha)$ takes in each of them.
4. d) Based on your study, determine whether it is possible to find any angle $\alpha$ such that $E(\alpha) = 2$ and describe what position that angle occupies on the unit circle.