Trigonometric Functions

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

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

• $f(x) = x^2 \sin(x)$

• $g(x) = \cos(x) + x^4$

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

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

• $x = \frac{3\pi}{4}$

• $x = \frac{5\pi}{3}$

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

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

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

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 $4\pi$.

The height $h$ (in meters) 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 $C$.

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.

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 $\frac{\pi}{2}$.

The depth of the water $d$ (in meters) at a harbor entrance varies with the tides and can be modeled 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 harbor?

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

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.

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?

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 ($t=80$).

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.

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

• Determine its domain.

• Determine its range.

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

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 $50\sqrt{3}\text{ m}$.

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).

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?