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 $x = \frac{3\pi}{2}$.

Observe the following graph and determine whether it corresponds to the sine function or the cosine function. Justify your answer by indicating the y-axis intercept and the type of symmetry it presents.

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

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

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

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

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

The height in feet of one of the cabins on a Ferris wheel in terms 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 Ferris 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 feet over $24\text{ hours}$. Determine the amplitude, the period and the mean height of the water. From this data, propose a trigonometric expression of the type $h(t) = A\sin(Bt) + D$ 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 household electrical system can be expressed by the function $V(t) = 325 \sin(100\pi t)$, where $V$ is the voltage in volts and $t$ the time in seconds. Calculate the maximum voltage reached and find out how many times per second the voltage takes exactly the value of $0\text{ voltios}$.

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

The number of hours of sunlight in a certain city throughout the year can be approximated by 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 January 1). How many hours of daylight are there exactly on day $d=80$? And on day $d=262$ (half a year later)? Interpret the meaning of both results.

Draw a sketch 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 $f(x) = \sin(x)$.

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 oscillates from side to side. Its horizontal displacement in inches from the center is modelled by the function $x(t) = A \cos(Bt)$. If it is known that the pendulum reaches a maximum displacement of $12\text{ in}$ and completes a full back-and-forth cycle every $2\text{ seconds}$, determine the exact values of the parameters $A$ and $B$. Furthermore, calculate at what exact instants during the first $4\text{ seconds}$ the pendulum passes right through the central equilibrium position.

The mean monthly temperature of a city is modelled with a trigonometric function. Observe the graph representing the temperature $T$ in degrees Fahrenheit in terms of the month $m$ (where $m=0$ corresponds to the beginning of January, $m=1$ to the beginning of 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 their emotional capacity a cycle of $28\text{ days}$. If both capacities are modelled with sine functions of amplitude $1$ starting from zero on the day of birth ($t=0$), write 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 passing to have positive values?

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 the time in months ($t=0$ is January 1). 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 of a rotating lighthouse illuminates a long, straight cliff. The distance $d$ (in feet) from the closest point of the cliff to the illuminated point at the instant $t$ (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 light beam when $t$ approaches $2\text{ seconds}$.

Prove 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 resting person, measured in liters, is modelled by 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 (complete cycles of inhalation and exhalation) this person takes in exactly one minute.