Exercises: Trigonometric function

Determine the domain and the 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 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 with 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 with respect to the basic sine function.

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

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

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

The depth of the water $d$ (in feet) at the entrance to a port varies with the tides and can be modelled by 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 is the maximum and minimum depth of the water in the port?

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

Find the analytical expression of 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 instant $t > 0$ at which the voltage reaches its maximum value of $311$ V.

A tracking camera is located at $50\text{ m}$ 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 vertically oscillating spring is modelled by the function $y(t) = -5\cos(4\pi t)$, where $y$ is the displacement in inches with respect to the equilibrium position ($y=0$) and $t$ is the time in seconds.

• What is the amplitude and the period of the motion?

• At what instant $t > 0$ 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.

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

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

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

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

Given the functions $f(x) = \sin(2x)$ and $g(x) = \sin(x)$ represented in the attached graph in the interval $[0, 2\pi]$.

Calculate analytically the exact coordinates of all the points of intersection between both curves in the interval $[0, 2\pi]$. (Hint: use the double angle identity for sine).

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

• Prove analytically whether the 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 the phase shift of the function.

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

Let $\theta$ be the angle formed by the radius drawn to an upper vertex of the rectangle and 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?