Exercises: Trigonometric functions

Determine the domain and range of the trigonometric function $f(x) = -4\cos(x) + 2$.

Calculate the period of the function $g(t) = \sin\left(\frac{\pi}{3}t\right)$.

Find the x-intercepts (points of intersection with the x-axis) of the function $h(x) = 2\sin(x) - 1$ in the interval $[0, 2\pi]$.

Study the symmetry (even, odd or neither) of the following trigonometric functions:

• $f(x) = x^2 \sin(x)$

• $g(x) = \cos(x) + x^4$

Evaluate the function $f(x) = \tan(x)$ at the following exact values expressed in radians:

• $x = \frac{3\pi}{4}$

• $x = \frac{5\pi}{3}$

Calculate the exact value in radians of the following expressions involving inverse trigonometric functions, without using a calculator:

• $\arcsin\left(-\frac{\sqrt{2}}{2}\right)$

• $\arccos(0)$

• $\arctan(\sqrt{3})$

Given the function $y = 5 - 3\sin(2x)$, identify its amplitude and its vertical translation relative to the basic sine function.

Write the algebraic expression for a function of the type $f(x) = A\cos(Bx)$, knowing that its amplitude is $5$ and its period is $4\pi$.

When studying the function $f(x) = -3\cos(2x) + 1$, Lucas states in his notebook:

"Since the leading coefficient is $-3$, the amplitude of the oscillation is $-3$. Furthermore, since $x$ is multiplied by $2$, the function completes its cycles twice as slowly, so the period is $T = 2 \cdot 2\pi = 4\pi$. Finally, the range goes from $-3 + 1$ to $3 + 1$, that is, $[-2, 4]$".

1. a) Reasonably identify the two conceptual errors made by Lucas in his explanation.
2. b) Explain why the range he obtained is numerically correct despite having reasoned the amplitude incorrectly.
3. c) Write the true, justified values of the amplitude and the period of $f(x)$.

Write the analytical expression of a trigonometric function of the form $f(x) = A\cos(Bx) + D$ (with $A > 0$ and $B > 0$) that simultaneously satisfies the following three conditions:

1. Its range or image is the interval $[-1, 7]$.

2. Its fundamental period is $4$ units.

3. At $x = 0$ the function reaches one of its absolute maxima.

Justify step by step how you deduce in a reasoned way the value of each of the parameters $A$, $B$, and $D$ from the given geometric properties.

The height $h$ (in metres) of a Ferris wheel capsule above the ground as a function of time $t$ (in minutes) is given by the attached graph.

From the graph, determine the algebraic expression for the function in the form $h(t) = A\cos(Bt) + C$. Justify the values of the parameters $A$, $B$ and $C$.

The depth of the water $d$ (in metres) at a harbour entrance varies with the tides and can be modelled using the function $d(t) = 12 + 4\sin\left(\frac{\pi}{6}t\right)$, where $t$ is the time in hours elapsed since midnight ($t=0$).

• What are the maximum and minimum depths of the water in the harbour?

• At what times of day is the maximum depth reached for the first and second time?

Find the algebraic expression for the function $f(x) = A\sin(Bx) + C$, knowing that:

• Its maximum value is $4$ and its minimum value is $-2$.

• Its period is $\pi$.

• It passes through the point $(0, 1)$ and is increasing at that instant.

Determine the domain of the function $f(x) = \frac{1}{2\cos(x) - 1}$ in the interval $[0, 2\pi]$.

The voltage $V$ (in volts) of an alternating current circuit as a function of time $t$ (in seconds) is given by $V(t) = 311\sin(100\pi t)$.

• Calculate the frequency of the current (number of complete cycles per second).

• Determine the first time $t > 0$ at which the voltage reaches its maximum value of $311$ V.

A tracking camera is located $50\text{ m}$ from the launch pad of a rocket. The rocket ascends vertically.

• Express the angle of elevation $\theta$ of the camera as an inverse trigonometric function of the height $h$ of the rocket.

• Calculate the exact angle of elevation (in radians and degrees) when the rocket is at a height of $50\sqrt{3}\text{ m}$.

The position of a weight suspended from a spring oscillating vertically is modelled using the function $y(t) = -5\cos(4\pi t)$, where $y$ is the displacement in cm from the equilibrium position ($y=0$) and $t$ is the time in seconds.

• What are the amplitude and period of the motion?

• At what time $t > 0$ does the weight pass through the equilibrium position for the first time?

Given the function $f(x) = 2\arcsin\left(\frac{x}{3}\right) + \pi$:

• Determine its domain.

• Determine its range.

On a spring day in Córdoba, the outdoor temperature $T$ (in $^{\circ}\text{C}$) over an entire day is reasonably approximated by the model:

where $t$ is the time in hours elapsed since midnight ($t = 0$).

1. a) Without performing analytical calculations using the formula, intuitively estimate at what time of day you would expect the minimum and maximum temperatures, and what temperature value you would expect to have around $12:00$ noon and around $18:00$.
2. b) Now calculate, using the analytical expression, the exact values of the maximum temperature, the minimum temperature, and the temperature at $12:00$ ($t = 12$) and at $18:00$ ($t = 18$).
3. c) Compare your previous estimates with the results obtained from the model: do your expectations match the graph of a reflected and translated cosine? Justify the consistency of the physical result with the mathematical expression.

To calculate all the vertical asymptotes of the function $f(x) = \tan\left(2x - \frac{\pi}{4}\right)$ in the closed interval $[0, \pi]$, two students propose two different strategies:

• Method A (analytical): State the condition for the tangent being undefined by equating the argument to the general prohibited values, $2x - \frac{\pi}{4} = \frac{\pi}{2} + k\pi$ with $k \in \mathbb{Z}$, solve for $x$, and select the integer values of $k$ that give solutions within the interval $[0, \pi]$.

• Method B (geometric via transformations): Take the standard asymptotes of the basic tangent $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$, apply to them first the horizontal phase shift (shift to the right) and then the horizontal scale compression (factor $1/2$), and from there add the new period.

1. a) Apply both methods detailing the intermediate steps and verify that both lead to the same asymptote lines in $[0, \pi]$.
2. b) Compare both approaches: which one do you consider more reliable and less prone to missing solutions if the required interval were much wider, for example $[0, 10\pi]$? Justify your choice.

In a city, the number of daylight hours throughout the year can be approximated by a function of the form $L(t) = A\cos\left(\frac{2\pi}{365}(t - C)\right) + D$, where $t$ is the day of the year ($t=1$ is 1 January).

Given that the day with the most daylight hours is 21 June ($t=172$) with $15$ hours, and the day with the fewest daylight hours is 21 December ($t=355$) with $9$ hours:

• Find the values of the parameters $A$, $C$ and $D$.

• Estimate the number of daylight hours on 21 March ($t=80$).

Given the functions $f(x) = \sin(2x)$ and $g(x) = \sin(x)$, analytically calculate the exact coordinates of all intersection points between both curves in the interval $[0, 2\pi]$. (Hint: use the double-angle sine identity).

Find the values of the parameters $a$, $b$ and $c$ (with $a > 0$ and $b > 0$) of the function $f(x) = a\sin(bx) + c$ knowing that:

• The maximum value of the function is $10$.

• The minimum value of the function is $2$.

• The period of the function is $\frac{\pi}{2}$.

Determine the domain of the function $f(x) = \sqrt{\cos(x)} + \ln(\sin(x))$ restricted to the interval $[0, 2\pi]$.

Solve analytically the inverse trigonometric equation: $\arcsin(x) = \arccos(x)$.

The monthly sales $S$ (in thousands of units) of a seasonal product are modelled by the function $S(t) = 120 + 20t + 50\cos\left(\frac{\pi}{6}t\right)$, where $t$ is the time in months ($t=0$ corresponds to January).

• Calculate sales in January ($t=0$) and in July ($t=6$).

• Prove analytically whether sales in July are greater or less than in January and calculate the exact difference.

Express the function $f(x) = \sqrt{3}\sin(x) + \cos(x)$ in the form $f(x) = A\sin(x + \alpha)$. From this new expression, state the amplitude and phase shift of the function.

A rectangle is inscribed inside a semicircle of radius $R = 10\text{ cm}$, such that its base lies on the diameter of the semicircle, as shown in the figure.

Let $\theta$ be the angle formed by the radius drawn to a top vertex of the rectangle with the diameter.

• Express the area of the rectangle as a function of the angle $\theta$. (Hint: use the double-angle formula for sine).

• For what value of $\theta \in \left(0, \frac{\pi}{2}\right)$ is the area of the rectangle maximum, and what is this maximum value?

In the port of Santander, the phenomenon of tides causes a periodic oscillation of the water level. On a specific day of spring tides, the following official high and low tide data are recorded:

• A low tide (minimum water level) occurs at 03:00 with an elevation of $0,8\text{ m}$.

• The next high tide (maximum water level) occurs at 09:12 ($t = 9,2\text{ h}$) reaching $4,6\text{ m}$.

• The tidal cycle is semidiurnal and has a full period of $12,4\text{ hours}$.

1. a) Construct in a reasoned way a function of the form $h(t) = A\cos(B(t - C)) + D$ (with $A > 0$) that models the water height in meters as a function of time $t$ (in hours elapsed since midnight, $t=0$). Justify the analytical deduction of the four parameters $A$, $B$, $C$, and $D$.
2. b) A merchant ship needs a minimum water draft of $3,65\text{ m}$ to enter the harbor entrance without running aground. State the corresponding trigonometric inequality and algebraically determine the morning time window (between 03:00 and 15:00) during which access to the port is safe for this ship.

In an assessment test, students are asked to simplify the composite function $g(x) = \arccos(\cos x)$ and calculate its exact value for $x = \frac{7\pi}{4}$. Marta writes on her exam:

"Since the arccosine is the inverse function of cosine, both functions directly cancel each other out, so $\arccos(\cos x) = x$ for any real number $x$. Consequently, $\arccos\left(\cos\left(\frac{7\pi}{4}\right)\right) = \frac{7\pi}{4}$".

1. a) Explain rigorously why Marta's general reasoning is false. In your explanation, you must explicitly refer to the lack of injectivity of the cosine function on $\mathbb{R}$ and to how the range of the arccosine function is formally defined.
2. b) Find the exact numerical value of $\arccos\left(\cos\left(\frac{7\pi}{4}\right)\right)$, justifying the result using the unit circle or the symmetry properties of trigonometric ratios.
3. c) Deduce a simplified formula for $g(x) = \arccos(\cos x)$ that is valid specifically when $x$ belongs to the interval $[\pi, 2\pi]$.

Unit challenge. "The daily energy pulse". You are going to investigate the relationship between weather and consumption in your region to propose efficiency measures in your local environment.

1. Data collection: Access the websites of REE (Red Eléctrica de España) and AEMET. Record hourly data ($24$ hours of the same day) for electricity demand and temperature in your province in a spreadsheet.

2. Modelling in GeoGebra:

3. Model the thermal fluctuation using a trigonometric function $f(t) = A\cos(B(t-C)) + D$.

4. Fit the demand curve using a polynomial $P(t)$ and calculate its derivative function $P'(t)$ to locate the hours when consumption grows fastest.

5. Model HVAC performance according to outdoor temperature by formulating a rational function and a root-based function.

6. Represent the grid saturation response to a thermal peak applying exponential and logarithmic models.

7. Reasoned proposal: Prepare a brief executive report (maximum $2$ pages) with the overlaid graphs, justifying which type of function best describes each phenomenon and providing saving guidelines based on the detected asymptotes and extrema.

Assessment criteria:

• Rigour when importing real data and calibrating function parameters in GeoGebra.

• Correct use of the polynomial derivative and the essential characteristics of each function (periodicity, asymptotes, curvature).

• Critical capacity to select the optimal model and mathematically justify the proposals.