A Ferris wheel on a Chicago pier turns at a steady pace. Its cabins swing between 4 ft above the ground at the bottom and 100 ft at the top, and one full turn takes 8 minutes. How high is your cabin 1 minute after you board? When do you first reach 76 ft? One formula answers both questions. It uses the trigonometric functions, now measured in radians.
Sine and Cosine as Functions
Put a point on the unit circle at an angle of $x$ radians. Its horizontal coordinate is $\cos x$ and its vertical coordinate is $\sin x$. As $x$ grows, the point goes around the circle again and again, so both coordinates repeat. That repetition is what makes these functions periodic.
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Determine the domain and the range of the trigonometric function $f(x) = -4\cos(x) + 2$.
Determine the domain and the range of the trigonometric function $f(x) = -4\cos(x) + 2$.
To evaluate trig functions without a calculator, use the unit circle. The three special angles in the first quadrant ($\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$) give every other exact value by symmetry.
The labels in each quadrant show which of sine, cosine, and tangent are positive there. In Quadrant I, All three are positive. In II, only Sine. In III, only Tangent. In IV, only Cosine. A common mnemonic is "All Students Take Calculus."
$\theta$
$0$
$\dfrac{\pi}{6}$
$\dfrac{\pi}{4}$
$\dfrac{\pi}{3}$
$\dfrac{\pi}{2}$
$\sin\theta$
$0$
$\dfrac12$
$\dfrac{\sqrt2}{2}$
$\dfrac{\sqrt3}{2}$
$1$
$\cos\theta$
$1$
$\dfrac{\sqrt3}{2}$
$\dfrac{\sqrt2}{2}$
$\dfrac12$
$0$
$\tan\theta$
$0$
$\dfrac{\sqrt3}{3}$
$1$
$\sqrt3$
undefined
When a problem asks for the x-intercepts (zeros) of a trig function, set the function equal to $0$, isolate $\sin x$ or $\cos x$, and use the reference angle. Usually there are two solutions in one period.
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Find the points of intersection with the x-axis of the function $h(x) = 2\sin(x) - 1$ in the interval $[0, 2\pi]$.
Find the points of intersection with the x-axis of the function $h(x) = 2\sin(x) - 1$ in the interval $[0, 2\pi]$.
The tangent is a quotient, $\tan x = \dfrac{\sin x}{\cos x}$, so it is undefined wherever $\cos x = 0$. Near those points the graph shoots off to $\pm\infty$ and has vertical asymptotes.
Any denominator can remove points
The same idea works for any expression with trig functions in a denominator. Set the denominator equal to $0$, solve in the given interval, and exclude those values.
Square roots and logarithms
A square root needs its radicand to be $\ge 0$. A logarithm needs its argument to be $> 0$. For a function with several pieces, write one condition for each piece and then find where all conditions hold at once.
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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 $f(x) = \frac{1}{2\cos(x) - 1}$ in the interval $[0, 2\pi]$.
Every sine or cosine wave is the basic wave stretched, squeezed, and moved. This is the same set of transformations you know from other functions. Here they have special names.
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Calculate the period of the function $g(t) = \sin\left(\frac{\pi}{3}t\right)$.
Calculate the period of the function $g(t) = \sin\left(\frac{\pi}{3}t\right)$.
Often you are told the maximum, minimum, and period and asked for the formula. Follow the same recipe every time.
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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…
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.
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…
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:
Modeling Periodic Motion: Ferris Wheels, Tides, and Circuits
Anything that repeats at a steady pace can be modeled by a sinusoid, where $t$ is usually time. To read a model from a graph, find the highest and lowest points for the max and min, and measure from peak to peak for the period. Then use the recipe from the last section.
Back to the start: one minute after boarding you are about $18.1$ ft up. You first reach $76$ ft after $\dfrac{8}{3}\approx 2.67$ minutes, and the top ($100$ ft) after exactly $4$ minutes.
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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…
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 modeled by the function $d(t) = 12 + 4\sin\left(\frac{\pi}{6}t\right)$,…
The depth of the water $d$ (in feet) at the entrance to a port varies with the tides and can be modeled 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?
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 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.
Some cycles do not start at a max or min when $t=0$. The usual trick is to use $\cos$ and set $C$ equal to a time when you know the maximum happens. Then $f(t)=A\cos\big(B(t-C)\big)+D$ starts its peak at $t=C$.
A cycle plus a trend
Real data often combine a cycle with growth, like $S(t)=100+10t+30\cos\left(\dfrac{\pi}{3}t\right)$. The cosine part oscillates and the $10t$ part keeps rising, so the whole function is not periodic. You can still compare values exactly if you pick inputs where the cosine is a known value.
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The position of a weight suspended from a vertically oscillating spring is modeled by the function $y(t) = -5\cos(4\pi t)$, where $y$ is the displacement in inches with…
The position of a weight suspended from a vertically oscillating spring is modeled 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?
In a city, the number of hours of sunlight throughout the year can be approximated by a function of the type $L(t) = A\cos\left(\frac{2\pi}{365}(t - C)\right) + D$,…
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 ($t=80$).
The monthly sales $S$ (in thousands of units) of a seasonal product are modeled by the function $S(t) = 120 + 20t + 50\cos\left(\frac{\pi}{6}t\right)$, where $t$ is the…
The monthly sales $S$ (in thousands of units) of a seasonal product are modeled 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.
Sine, cosine, and tangent are not one-to-one on all of $\mathbb{R}$, because they repeat their values. To get inverse functions we restrict the domain to one piece where each function is one-to-one.
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Calculate the exact value in radians of the following expressions with inverse trigonometric functions, without using a calculator:…
Calculate the exact value in radians of the following expressions with inverse trigonometric functions, without using a calculator:
Trig functions also describe lengths and areas that change with an angle.
Angles of elevation
If a camera is $d$ feet from the base of something that rises vertically, and the object is at height $h$, then $\tan\theta=\dfrac{h}{d}$. To get the angle from the height, use the inverse: $\theta=\arctan\dfrac{h}{d}$. This is valid because $\theta$ is acute, inside the range of $\arctan$.
Areas that depend on an angle
When a shape is inscribed in a circle, put a corner at the point $(R\cos\theta,\ R\sin\theta)$ on the circle. Then the side lengths are products of $R$ with $\cos\theta$ or $\sin\theta$. The area becomes a trig function whose maximum you can read from the range of sine or cosine.
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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…
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 $50\sqrt{3}\text{ m}$.
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 $\theta$…
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?
Two more tools show up when trig functions are mixed together.
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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…
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).
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…
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.