Exercises: Trigonometric functions: sine, cosine and tangent.

Determine the amplitude and period of the trigonometric function $f(x) = 4\sin(3x)$.

Evaluate the function $g(x) = 2\cos(x) - 1$ at the values $x = 0$, $x = \frac{\pi}{2}$, $x = \pi$ and $x = \frac{3\pi}{2}$.

Look at the following graph and determine whether it corresponds to the sine function or the cosine function. Justify your answer by stating the y-intercept and the type of symmetry it exhibits.

State the domain and range of the trigonometric function $h(x) = 5\sin(x) + 2$.

Classify the basic functions $f(x) = \sin(x)$, $g(x) = \cos(x)$ and $h(x) = \tan(x)$ according to the type of symmetry they exhibit (even or odd).

Calculate the exact period of the function $y = \tan\left(\frac{x}{3}\right)$.

Find all x-intercepts of the function $y = \sin(x)$ in the closed interval $[0, 2\pi]$.

Describe the geometric transformation undergone by the graph of the function $f(x) = \cos(x)$ to become the graph of $g(x) = \cos(x) - 3$. What is the new range of the function $g(x)$?

The height in meters of one of the capsules on a Ferris wheel as a function of time in minutes is given by the expression $h(t) = 20 - 15\cos\left(\frac{\pi}{2}t\right)$. Calculate the maximum and minimum height reached by the wheel, and determine the exact time it takes to complete one full revolution.

The water level in a port varies periodically due to the tides. The following graph shows the height of the water in meters over $24\text{ hours}$. Determine the amplitude, period, and mean water height. From these data, propose a trigonometric expression of the form $h(t) = A\sin(Bt) + D$ that fits the graph shown.

A pure sound wave emitted by a tuning fork is modelled by the function $P(t) = 0.02 \sin(880\pi t)$, where $P$ represents the pressure variation in pascals and $t$ the time elapsed in seconds. Determine the amplitude of the wave and its frequency (the number of complete cycles it performs per second).

The voltage of an alternating current in a European domestic installation can be expressed using the function $V(t) = 325 \sin(100\pi t)$, where $V$ is the voltage in volts and $t$ is the time in seconds. Calculate the maximum voltage reached and find out how many times per second the voltage takes the value of $0\text{ voltios}$ exactly.

A weight hanging from the end of a spring oscillates vertically. Its position relative to the ceiling is given by the equation $y(t) = 8\cos(4t) - 20$, measured in centimeters. State the distance of the weight from the ceiling at the initial moment ($t=0$), what its lowest position (greatest distance from the ceiling) is, and what the period of oscillation of the spring is.

The number of hours of daylight in a given city throughout the year can be approximated using the function $H(d) = 12 + 3\sin\left(\frac{2\pi}{365}(d-80)\right)$, where $d$ represents the day of the year ($d=1$ is 1 January). How many hours of daylight are there on day $d=80$ exactly? And on day $d=262$ (half a year later)? Interpret the meaning of both results.

Sketch the graph of the function $g(x) = 3\sin\left(x - \frac{\pi}{2}\right)$. Explain step by step how this graph is obtained by applying translations and stretches to the original graph of the function $f(x) = \sin(x)$.

You can do it on paper and send a photo with your answer.

Determine the domain of the function $f(x) = \tan(2x)$ and write the equations of its first three positive vertical asymptotes.

The pendulum of a wall clock swings from side to side. Its horizontal displacement in centimeters from the center is modeled by the function $x(t) = A \cos(Bt)$. Knowing that the pendulum reaches a maximum displacement of $12\text{ cm}$ and completes a full back-and-forth cycle every $2\text{ seconds}$, determine the exact values of the parameters $A$ and $B$. Furthermore, calculate the exact times during the first $4\text{ seconds}$ when the pendulum passes right through the central equilibrium position.

The monthly mean temperature of a city is modelled using a trigonometric function. Observe the graph representing the temperature $T$ in degrees Celsius as a function of the month $m$ (where $m=0$ corresponds to early January, $m=1$ to early February, etc.). Determine the values of the parameters $A, B, C$ and $D$ so that the function $T(m) = A \cos(B(m - C)) + D$ accurately represents this curve.

Find analytically all the points of intersection ($x$ and $y$ coordinates) of the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ in the closed interval $[0, 2\pi]$. Justify your answer using the properties of trigonometric functions.

According to biorhythm theory, a person's physical capacity follows a sinusoidal cycle of $23\text{ days}$ and emotional capacity a cycle of $28\text{ days}$. If both capacities are modelled by sine functions of amplitude $1$ starting from zero on the day of birth ($t=0$), write down the functions $F(t)$ and $E(t)$. How many exact days must pass from birth for both capacities to cross the zero value simultaneously again, with both becoming positive?

The rabbit population in a nature reserve fluctuates annually according to the model $P(t) = 800 + 300\sin\left(\frac{\pi t}{6}\right)$, where $t$ represents time in months ($t=0$ is 1 January). Set up and solve the corresponding trigonometric inequality to determine exactly during which months of the first year the population remains strictly above $950\text{ conejos}$.

The light beam from a rotating lighthouse illuminates a long straight cliff. The distance $d$ (in meters) from the closest point on the cliff to the illuminated spot at time $t$ (in seconds) is given by the function $d(t) = 200 \tan\left(\frac{\pi}{4}t\right)$. Determine the domain of this function in the physical context of the problem (for the first rotation) and interpret what happens to the beam of light as $t$ approaches $2\text{ seconds}$.

Show, by analysing horizontal translations, that the function $f(x) = \sin\left(x + \frac{\pi}{2}\right)$ is equivalent to the function $g(x) = \cos(x)$. Use the concepts of phase shift, periodicity and the shape of their graphs to justify your answer.

The volume of air present in the lungs of a person at rest, measured in liters, is modeled using the function $V(t) = 2.5 + 0.5\cos\left(\frac{\pi}{2}t\right)$, where $t$ is measured in seconds. Calculate the maximum and minimum volume of air in the lungs. From the period of the function, determine how many complete breaths (full cycles of inhalation and exhalative phase) this person takes in exactly one minute.